A Prime Game:

Write down a multidigit prime number (i.e. a prime number > 10), and I can always strike out 0 or more digits to get a prime in this list:

{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

e.g.

* Write down the prime 149 → I can strike out the digit 4, to get the prime 19
* Write down the prime 439 → I can strike out the digit 9, to get the prime 43
* Write down the prime 857 → I can strike out zero digits, to get the prime 857
* Write down the prime 2081 → I can strike out the digit 0, to get the prime 281
* Write down the largest known double Mersenne prime 170141183460469231731687303715884105727 (2<sup>127</sup>−1) → I can strike out all digits except the third-leftmost 1 and the second-rightmost 3, to get the prime 13 (also I can choose to strike out all digits except the second-leftmost 4 and the third-rightmost 7, to get the prime 47)
* Write down the largest known Fermat prime 65537 → I can strike out the 6 and the 3, to get the prime 557 (also I can choose to strike out the 6 and two 5's, to get the prime 37) (also I can choose to strike out two 5's and the 3, to get the prime 67) (also I can choose to strike out the 6, one 5, and the 7, to get the prime 53)
* Write down the famous repunit prime 1111111111111111111 (with 19 1's) → I can strike out 17 1's, to get the prime 11
* Write down the prime 1000000000000000000000000000000000000000000000000000000000007 (which is the next prime after 10<sup>60</sup>) → I can strike out all 0's, to get the prime 17
* Write down the prime 95801 → I can strike out the 9, to get the prime 5801
* Write down the prime 946969 → I can strike out the first 9 and two 6's, to get the prime 499
* Write down the prime 90000000581 → I can strike out five 0's, the 5, and the 8, to get the prime 9001
* Write down the prime 8555555555555555555551 → I can strike out the 8 and nine 5's, to get the prime 555555555551

These primes are called **minimal primes**.

Let *b* > 1 be a natural number (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html). A minimal prime base *b* is a prime number (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, https://oeis.org/A000040) greater than (https://en.wikipedia.org/wiki/Greater_than, https://mathworld.wolfram.com/Greater.html) *b* whose base-*b* (i.e. the positional numeral system (https://en.wikipedia.org/wiki/Positional_numeral_system) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b*) representation has no proper subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) which is also a prime number greater than *b*.

For example, 857 is a minimal prime in decimal (base *b* = 10) because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a minimal prime in decimal (base *b* = 10) (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a minimal prime in decimal (base *b* = 10) even though a subset of the digits can form the shorter prime 19 > 10 by changing the order.

Now we extend minimal primes to bases *b* other than 10.

The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) (https://mathworld.wolfram.com/MaximalElement.html for maximal element, the dual of minimal element, unfortunely there is no article "minimal element" in mathworld, a minimal element of a set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) under a partial ordering binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is a maximal element of the same set under its converse relation (https://en.wikipedia.org/wiki/Converse_relation), a converse relation of a partial ordering relation must also be a partial ordering relation) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, https://oeis.org/A000040) which are > *b* written in the positional numeral system (https://en.wikipedia.org/wiki/Positional_numeral_system) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b*, as digit (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html) strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), for 2 ≤ *b* ≤ 36 (I stop at base 36 since this base is a maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet)), references: http://www.tonymarston.net/php-mysql/converter.html, https://www.dcode.fr/base-36-cipher, http://www.urticator.net/essay/5/567.html, http://www.urticator.net/essay/6/624.html, https://docs.python.org/3/library/functions.html#int, https://reference.wolfram.com/language/ref/BaseForm.html, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36), using A−Z to represent digit values 10 to 35.

A string (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) *x* is a subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) of another string *y*, if *x* can be obtained from *y* by deleting zero or more of the characters (https://en.wikipedia.org/wiki/Character_(computing)) (in this project, digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)) in *y*. For example, 514 is a subsequence of 352148, "STRING" is a subsequence of "MEISTERSINGER". In contrast, 758 is not a subsequence of 378259, "ABC" is not a subsequence of "CBACACBA", since the characters (in this project, digits) must be in the same order. The empty string (https://en.wikipedia.org/wiki/Empty_string) *𝜆* is a subsequence of every string. There are 2<sup>*n*</sup> subsequences of a string with length *n*, e.g. the subsequences of 123456 are (totally 2<sup>6</sup> = 64 subsequences):

*𝜆*, 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56, 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456, 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 12345, 12346, 12356, 12456, 13456, 23456, 123456

"The set of strings ordered by subsequence" is a partially ordered set (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), since this binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is reflexive (https://en.wikipedia.org/wiki/Reflexive_relation, https://mathworld.wolfram.com/Reflexive.html), antisymmetric (https://en.wikipedia.org/wiki/Antisymmetric_relation), and transitive (https://en.wikipedia.org/wiki/Transitive_relation), and thus we can draw its Hasse diagram (https://en.wikipedia.org/wiki/Hasse_diagram, https://mathworld.wolfram.com/HasseDiagram.html) and find its greatest element (https://en.wikipedia.org/wiki/Greatest_element), least element (https://en.wikipedia.org/wiki/Least_element), maximal elements (https://en.wikipedia.org/wiki/Maximal_element, https://mathworld.wolfram.com/MaximalElement.html), and minimal elements (https://en.wikipedia.org/wiki/Minimal_element), however, the greatest element and least element may not exist, and for an infinite set (such as the set of the "prime numbers > *b*" strings in base *b* (for a given base *b* ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html)), the maximal elements also may not exist, thus we are only interested on finding the minimal elements of these sets, and we define "minimal set" of a set as the set of the minimal elements of this set, under a given partially ordered binary relation (this binary relation is "is a subsequence of" in this project))

By the theorem that there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/InfiniteSet.html) antichains (https://en.wikipedia.org/wiki/Antichain, https://mathworld.wolfram.com/Antichain.html) (i.e. a subset of a partially ordered set such that any two distinct elements in the subset are incomparable (https://en.wikipedia.org/wiki/Comparability, https://mathworld.wolfram.com/ComparableElements.html)) for the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) (i.e. the set of the minimal elements of any set under the subsequence ordering must be finite, even if this set is infinite, such as the set of the "prime numbers > *b*" strings in base *b* (for a given base *b* ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html), there must be only finitely such minimal elements in every base *b*.

In this project, we will find the sets of the minimal elements of these 35 sets under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html):

|*b*|the base *b* representations of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, https://oeis.org/A000040) > *b* (only list the first 500 elements in the sets)|
|---|---|
|2|11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 111011, 111101, 1000011, 1000111, 1001001, 1001111, 1010011, 1011001, 1100001, 1100101, 1100111, 1101011, 1101101, 1110001, 1111111, 10000011, 10001001, 10001011, 10010101, 10010111, 10011101, 10100011, 10100111, 10101101, 10110011, 10110101, 10111111, 11000001, 11000101, 11000111, 11010011, 11011111, 11100011, 11100101, 11101001, 11101111, 11110001, 11111011, 100000001, 100000111, 100001101, 100001111, 100010101, 100011001, 100011011, 100100101, 100110011, 100110111, 100111001, 100111101, 101001011, 101010001, 101011011, 101011101, 101100001, 101100111, 101101111, 101110101, 101111011, 101111111, 110000101, 110001101, 110010001, 110011001, 110100011, 110100101, 110101111, 110110001, 110110111, 110111011, 111000001, 111001001, 111001101, 111001111, 111010011, 111011111, 111100111, 111101011, 111110011, 111110111, 111111101, 1000001001, 1000001011, 1000011101, 1000100011, 1000101101, 1000110011, 1000111001, 1000111011, 1001000001, 1001001011, 1001010001, 1001010111, 1001011001, 1001011111, 1001100101, 1001101001, 1001101011, 1001110111, 1010000001, 1010000011, 1010000111, 1010001101, 1010010011, 1010010101, 1010100001, 1010100101, 1010101011, 1010110011, 1010111101, 1011000101, 1011001111, 1011010111, 1011011101, 1011100011, 1011100111, 1011101111, 1011110101, 1011111001, 1100000001, 1100000101, 1100010011, 1100011101, 1100101001, 1100101011, 1100110101, 1100110111, 1100111011, 1100111101, 1101000111, 1101010101, 1101011001, 1101011011, 1101011111, 1101101101, 1101110001, 1101110011, 1101110111, 1110001011, 1110001111, 1110010111, 1110100001, 1110101001, 1110101101, 1110110011, 1110111001, 1111000111, 1111001011, 1111010001, 1111010111, 1111011111, 1111100101, 1111110001, 1111110101, 1111111011, 1111111101, 10000000111, 10000001001, 10000001111, 10000011001, 10000011011, 10000100101, 10000100111, 10000101101, 10000111111, 10001000011, 10001000101, 10001001001, 10001001111, 10001010101, 10001011101, 10001100011, 10001101001, 10001111111, 10010000001, 10010001011, 10010010011, 10010011101, 10010100011, 10010101001, 10010110001, 10010111101, 10011000001, 10011000111, 10011001101, 10011001111, 10011010101, 10011100001, 10011101011, 10011111101, 10011111111, 10100000011, 10100001001, 10100001011, 10100010001, 10100010101, 10100010111, 10100011011, 10100100111, 10100101001, 10100101111, 10101010001, 10101010111, 10101011101, 10101100101, 10101110111, 10110000001, 10110001111, 10110010011, 10110010101, 10110011001, 10110011111, 10110100111, 10110101011, 10110101101, 10110110011, 10110111111, 10111001001, 10111001011, 10111001111, 10111010001, 10111010101, 10111011011, 10111100111, 10111110011, 10111111011, 11000000111, 11000001101, 11000010001, 11000010111, 11000011111, 11000100011, 11000101011, 11000101111, 11000111101, 11001000001, 11001000111, 11001001001, 11001001101, 11001010011, 11001010101, 11001011011, 11001100101, 11001111001, 11001111111, 11010000011, 11010000101, 11010011101, 11010100001, 11010100011, 11010101101, 11010111001, 11010111011, 11011000101, 11011001101, 11011010011, 11011011001, 11011011111, 11011110001, 11011110111, 11011111011, 11011111101, 11100001001, 11100010011, 11100011111, 11100100111, 11100110111, 11101000101, 11101001011, 11101001111, 11101010001, 11101010101, 11101010111, 11101100001, 11101101101, 11101110011, 11101111001, 11110001011, 11110001101, 11110011101, 11110011111, 11110110101, 11110111011, 11111000011, 11111001001, 11111001101, 11111001111, 11111010011, 11111011011, 11111100001, 11111101011, 11111101101, 11111110111, 100000000101, 100000001111, 100000010101, 100000100001, 100000100011, 100000100111, 100000101001, 100000110011, 100000111111, 100001000001, 100001010001, 100001010011, 100001011001, 100001011101, 100001011111, 100001101001, 100001110001, 100010000011, 100010011011, 100010011111, 100010100101, 100010101101, 100010111101, 100010111111, 100011000011, 100011001011, 100011011011, 100011011101, 100011100001, 100011101001, 100011101111, 100011110101, 100011111001, 100100000101, 100100000111, 100100011101, 100100100011, 100100100101, 100100101011, 100100101111, 100100110101, 100101000011, 100101001001, 100101001101, 100101001111, 100101010101, 100101011001, 100101011111, 100101101011, 100101110001, 100101110111, 100110000101, 100110001001, 100110001111, 100110011011, 100110100011, 100110101001, 100110101101, 100111000111, 100111011001, 100111100011, 100111101011, 100111101111, 100111110101, 100111110111, 100111111101, 101000010011, 101000011111, 101000100001, 101000110001, 101000111001, 101000111101, 101001001001, 101001010111, 101001100001, 101001100011, 101001100111, 101001101111, 101001110101, 101001111011, 101001111111, 101010000001, 101010000101, 101010001011, 101010010011, 101010010111, 101010011001, 101010011111, 101010101001, 101010101011, 101010110101, 101010111101, 101011000001, 101011001111, 101011011001, 101011100101, 101011100111, 101011101101, 101011110001, 101011110011, 101100000011, 101100010001, 101100010101, 101100011011, 101100100011, 101100101001, 101100101101, 101100111111, 101101000111, 101101010001, 101101010111, 101101011101, 101101100101, 101101101111, 101101111011, 101110001001, 101110001101, 101110010011, 101110011001, 101110011011, 101110110111, 101110111001, 101111000011, 101111001011, 101111001111, 101111011101, 101111100001, 101111101001, 101111110101, 101111111011, 110000000111, 110000001011, 110000010001, 110000100101, 110000101111, 110000110001, 110001000001, 110001011011, 110001011111, 110001100001, 110001101101, 110001110011, 110001110111, 110010000011, 110010001001, 110010010001, 110010010101, 110010011101, 110010110011, 110010110101, 110010111001, 110010111011, 110011000111, 110011100011, 110011100101, 110011101011, 110011110001, 110011110111, 110011111011, 110100000001, 110100000011, 110100001111, 110100010011, 110100011111, 110100100001, 110100101011, 110100101101, 110100111101, 110100111111, 110101001111, 110101010101, 110101101001, 110101111001, 110110000001, 110110000101, 110110000111, 110110001011, 110110001101, 110110100011, 110110101011, 110110110111, 110110111101, 110111000111, 110111001001, 110111001101, 110111010011, 110111010101, 110111011011, 110111100101, 110111100111, 110111110011, 110111111101, ...|
|3|12, 21, 102, 111, 122, 201, 212, 1002, 1011, 1101, 1112, 1121, 1202, 1222, 2012, 2021, 2111, 2122, 2201, 2221, 10002, 10022, 10121, 10202, 10211, 10222, 11001, 11012, 11201, 11212, 12002, 12011, 12112, 12121, 12211, 20001, 20012, 20102, 20122, 20201, 21002, 21011, 21022, 21101, 21211, 22021, 22102, 22111, 22122, 22212, 22221, 100022, 100112, 100202, 100222, 101001, 101021, 101102, 101111, 101212, 102101, 102112, 102121, 102202, 110021, 110111, 110212, 110221, 111002, 111022, 111121, 111211, 112001, 112012, 112102, 112201, 112212, 120011, 120112, 120121, 120222, 121001, 121021, 121102, 121122, 121221, 122002, 122011, 122022, 122202, 200001, 200012, 200111, 200122, 200212, 201022, 201101, 202001, 202021, 202122, 202212, 210002, 210011, 210101, 210202, 210222, 211012, 211021, 211111, 211201, 211212, 211221, 212101, 212202, 212211, 212222, 220012, 220102, 220111, 220221, 221002, 221022, 221121, 221222, 222021, 222122, 222221, 1000011, 1000101, 1000112, 1000211, 1001001, 1001012, 1001111, 1001122, 1002011, 1002112, 1002222, 1010001, 1010102, 1010111, 1010122, 1010201, 1011002, 1011121, 1011202, 1011211, 1011222, 1012111, 1012122, 1012201, 1012212, 1020121, 1020202, 1021001, 1021102, 1021201, 1021212, 1022002, 1022022, 1022211, 1022222, 1100012, 1100102, 1100201, 1100221, 1101101, 1101112, 1101202, 1101211, 1102012, 1102021, 1102111, 1102212, 1102221, 1110022, 1110101, 1110121, 1111021, 1111102, 1111111, 1111122, 1111212, 1112002, 1112101, 1112121, 1112211, 1120122, 1120201, 1121002, 1121101, 1121202, 1121222, 1122012, 1122111, 1122221, 1200002, 1200022, 1200112, 1200121, 1200211, 1201021, 1201122, 1202022, 1202101, 1202112, 1202202, 1202211, 1210001, 1210012, 1210021, 1210102, 1210212, 1210221, 1211011, 1212102, 1212122, 1212212, 1220011, 1220211, 1221012, 1221201, 1221212, 1221221, 1222002, 1222022, 1222121, 1222202, 1222211, 2000001, 2000111, 2000212, 2000221, 2001002, 2001011, 2001022, 2001112, 2001222, 2002102, 2002201, 2010011, 2010101, 2010112, 2010202, 2011001, 2011012, 2011111, 2011122, 2012011, 2012022, 2012112, 2012121, 2012202, 2012222, 2020001, 2020021, 2020122, 2021101, 2021121, 2021202, 2021211, 2022201, 2022212, 2022221, 2100022, 2100202, 2100211, 2101012, 2101111, 2101201, 2101221, 2102011, 2102211, 2110001, 2110012, 2110021, 2110201, 2111002, 2111112, 2111211, 2112102, 2112221, 2120011, 2120022, 2120101, 2120112, 2120121, 2120222, 2121102, 2121122, 2121212, 2122112, 2122121, 2200012, 2200021, 2201002, 2201022, 2201121, 2201211, 2201222, 2202001, 2202012, 2202111, 2202201, 2210002, 2210011, 2210112, 2211001, 2211102, 2211122, 2212002, 2212011, 2212022, 2212101, 2212202, 2220012, 2220021, 2220212, 2220221, 2221011, 2221022, 2221101, 2221202, 2222001, 2222201, 10000121, 10000202, 10000222, 10001021, 10001212, 10001221, 10002002, 10002101, 10002222, 10010001, 10010012, 10010111, 10010201, 10010221, 10011002, 10011112, 10011121, 10012102, 10012122, 10012201, 10012221, 10020002, 10020022, 10020211, 10021001, 10021012, 10021021, 10021111, 10021122, 10021212, 10022022, 10022112, 10022202, 10100021, 10100102, 10100122, 10101002, 10101101, 10101121, 10101202, 10102201, 10110101, 10110202, 10111001, 10111012, 10111102, 10111111, 10111201, 10112112, 10112222, 10120001, 10120122, 10120221, 10121002, 10121112, 10122001, 10122102, 10122111, 10122122, 10122221, 10200011, 10200101, 10200112, 10200121, 10200202, 10200222, 10201021, 10201102, 10201111, 10201201, 10202002, 10202011, 10202112, 10202211, 10202222, 10210111, 10210212, 10211022, 10211101, 10211121, 10211202, 10211211, 10212102, 10212221, 10220002, 10220022, 10220121, 10220211, 10220222, 10221122, 10221221, 10222022, 10222112, 10222202, 11000001, 11000102, 11000212, 11001101, 11001112, 11001202, 11001222, 11002001, 11010002, 11010011, 11010112, 11010211, 11010222, 11011111, 11011122, 11011221, 11012101, 11012121, 11020001, 11020012, 11020102, 11021011, 11021112, 11021121, 11022012, 11100011, 11100022, 11100101, 11100211, 11101001, 11101012, 11101122, 11101212, 11102011, 11102022, 11102121, 11110102, 11110111, 11110122, 11110201, 11111011, 11112012, 11112021, 11112111, 11112201, 11112221, 11120002, 11120022, 11120101, 11120211, 11120222, 11121102, 11121111, 11121212, 11121221, 11122112, 11122121, 11200012, 11200102, 11201011, 11201202, 11202001, 11202012, 11202021, 11202102, 11202111, 11210022, 11210121, 11211001, 11211021, 11211122, 11211201, 11211212, 11212002, 11212011, 11212101, 11212202, 11212211, 11220021, 11220122, 11220201, ...|
|4|11, 13, 23, 31, 101, 103, 113, 131, 133, 211, 221, 223, 233, 311, 323, 331, 1003, 1013, 1021, 1033, 1103, 1121, 1201, 1211, 1213, 1223, 1231, 1301, 1333, 2003, 2021, 2023, 2111, 2113, 2131, 2203, 2213, 2231, 2303, 2311, 2333, 3001, 3011, 3013, 3103, 3133, 3203, 3211, 3221, 3233, 3301, 3323, 10001, 10013, 10031, 10033, 10111, 10121, 10123, 10211, 10303, 10313, 10321, 10331, 11023, 11101, 11123, 11131, 11201, 11213, 11233, 11311, 11323, 11333, 12011, 12031, 12101, 12121, 12203, 12211, 12233, 12301, 12313, 12323, 13001, 13021, 13031, 13033, 13103, 13133, 13213, 13223, 13303, 13313, 13331, 20021, 20023, 20131, 20203, 20231, 20303, 20321, 20323, 21001, 21023, 21101, 21113, 21121, 21133, 21211, 21221, 21223, 21313, 22001, 22003, 22013, 22031, 22103, 22111, 22201, 22211, 22223, 22303, 22331, 23011, 23033, 23113, 23131, 23203, 23213, 23233, 23311, 23321, 30001, 30011, 30103, 30131, 30221, 30223, 30311, 30313, 30323, 30331, 31013, 31111, 31121, 31123, 31133, 31231, 31301, 31303, 31313, 32023, 32033, 32113, 32201, 32221, 32231, 32303, 32321, 33013, 33023, 33101, 33113, 33133, 33211, 33301, 33311, 33323, 33331, 100013, 100021, 100033, 100121, 100123, 100211, 100213, 100231, 100333, 101003, 101011, 101021, 101033, 101111, 101131, 101203, 101221, 101333, 102001, 102023, 102103, 102131, 102203, 102221, 102301, 102331, 103001, 103013, 103031, 103033, 103111, 103201, 103223, 103331, 103333, 110003, 110021, 110023, 110101, 110111, 110113, 110123, 110213, 110221, 110233, 111101, 111113, 111131, 111211, 111313, 112001, 112033, 112103, 112111, 112121, 112133, 112213, 112223, 112231, 112303, 112333, 113021, 113023, 113033, 113101, 113111, 113123, 113213, 113303, 113323, 120013, 120031, 120101, 120113, 120133, 120203, 120223, 120233, 120331, 121001, 121013, 121021, 121031, 121103, 121111, 121123, 121211, 121321, 121333, 122003, 122011, 122131, 122201, 122203, 122231, 122321, 122323, 123011, 123031, 123103, 123121, 123133, 123301, 123313, 123323, 123331, 130021, 130103, 130133, 130213, 130313, 131011, 131023, 131033, 131101, 131111, 131113, 131201, 131231, 131303, 131321, 132023, 132031, 132131, 132133, 132311, 132323, 133003, 133021, 133031, 133033, 133103, 133123, 133201, 133223, 133231, 133313, 200011, 200033, 200111, 200201, 200203, 200213, 200221, 200303, 200333, 201001, 201101, 201103, 201121, 201131, 201133, 201221, 201301, 202003, 202123, 202133, 202211, 202231, 202331, 202333, 203003, 203023, 203123, 203131, 203201, 203221, 203233, 203311, 203321, 210011, 210013, 210131, 210203, 210211, 210223, 210233, 210311, 211003, 211021, 211031, 211033, 211111, 211121, 211133, 211223, 211301, 211313, 212011, 212021, 212033, 212123, 212203, 212221, 212231, 213013, 213121, 213203, 213223, 213233, 213311, 213313, 213331, 220103, 220133, 220201, 220301, 220321, 220331, 221021, 221113, 221201, 221203, 221213, 221233, 221311, 221323, 221333, 222001, 222011, 222023, 222103, 222113, 222121, 222133, 222221, 222223, 222311, 222331, 223001, 223033, 223121, 223211, 223213, 223231, 223301, 223303, 230003, 230101, 230111, 230123, 230203, 230221, 230231, 230333, 231013, 231101, 231113, 231131, 231211, 231233, 231323, 232021, 232031, 232103, 232121, 232123, 232313, 232321, 233003, 233023, 233033, 233131, 233201, 233221, 233311, 233323, 300013, 300023, 300101, 300211, 300233, 300301, 301001, 301123, 301133, 301201, 301231, 301303, 301313, 302003, 302021, 302101, 302111, 302131, 302303, 302311, 302321, 302323, 303013, 303203, 303211, 303223, 303301, 303313, 303323, 310001, 310003, 310033, 310103, 310133, 310201, 310223, 310231, 310331, 310333, 311033, 311111, 311221, 311321, 312001, 312011, 312013, 312023, 312031, 312203, 312223, 312313, 312331, 313013, 313021, 313031, 313103, 313111, 313123, 313211, 313213, 313303, 313331, 313333, ...|
|5|12, 21, 23, 32, 34, 43, 104, 111, 122, 131, 133, 142, 203, 214, 221, 232, 241, 243, 304, 313, 324, 342, 401, 403, 412, 414, 423, 1002, 1011, 1022, 1024, 1044, 1101, 1112, 1123, 1132, 1143, 1204, 1211, 1231, 1233, 1242, 1244, 1321, 1343, 1402, 1404, 1413, 1424, 1431, 2001, 2012, 2023, 2034, 2041, 2102, 2111, 2113, 2133, 2212, 2221, 2223, 2232, 2311, 2322, 2342, 2344, 2403, 2414, 2432, 2443, 3004, 3013, 3024, 3042, 3101, 3114, 3134, 3141, 3211, 3213, 3224, 3233, 3244, 3312, 3321, 3323, 3332, 3404, 3422, 3431, 3444, 4003, 4014, 4041, 4043, 4131, 4142, 4212, 4223, 4234, 4241, 4302, 4322, 4333, 4344, 4401, 4412, 4423, 4432, 4434, 10011, 10031, 10033, 10042, 10103, 10114, 10121, 10143, 10202, 10213, 10231, 10301, 10314, 10334, 10402, 10413, 10424, 10433, 11001, 11012, 11021, 11034, 11043, 11122, 11142, 11214, 11221, 11241, 11243, 11302, 11304, 11324, 11403, 11412, 11414, 11423, 12002, 12011, 12013, 12022, 12112, 12121, 12134, 12204, 12222, 12231, 12242, 12303, 12332, 12341, 12402, 12413, 12431, 12442, 13014, 13023, 13034, 13041, 13111, 13113, 13124, 13144, 13201, 13221, 13223, 13234, 13322, 13331, 13333, 13342, 13403, 13414, 13432, 13443, 14004, 14101, 14103, 14123, 14141, 14211, 14222, 14233, 14301, 14323, 14332, 14343, 14404, 14411, 14422, 14444, 20014, 20102, 20104, 20113, 20124, 20131, 20142, 20201, 20203, 20212, 20234, 20241, 20302, 20421, 20432, 20443, 21011, 21044, 21114, 21143, 21202, 21204, 21213, 21224, 21242, 21301, 21303, 21314, 21341, 21411, 21413, 21422, 21424, 21433, 21444, 22021, 22043, 22111, 22133, 22144, 22203, 22214, 22232, 22241, 22304, 22313, 22342, 22401, 22412, 22414, 22423, 22434, 22441, 23002, 23022, 23112, 23123, 23132, 23134, 23233, 23242, 23244, 23314, 23341, 23343, 23413, 23431, 23442, 24003, 24014, 24102, 24113, 24122, 24124, 24201, 24221, 24243, 24311, 24342, 24421, 24432, 24441, 24443, 30002, 30004, 30024, 30101, 30112, 30123, 30211, 30213, 30244, 30301, 30343, 30404, 30422, 30433, 30442, 30444, 31003, 31021, 31032, 31102, 31104, 31124, 31203, 31223, 31234, 31311, 31313, 31322, 31324, 31344, 31421, 31423, 32004, 32011, 32022, 32031, 32033, 32103, 32121, 32204, 32303, 32312, 32323, 32341, 32422, 32424, 32433, 33001, 33032, 33034, 33043, 33111, 33122, 33133, 33142, 33214, 33221, 33313, 33324, 33331, 33342, 33401, 33412, 33441, 34002, 34011, 34013, 34024, 34033, 34044, 34121, 34132, 34143, 34222, 34231, 34242, 34314, 34332, 34343, 34402, 40003, 40041, 40111, 40124, 40133, 40144, 40201, 40212, 40304, 40331, 40333, 40414, 40432, 40441, 41013, 41042, 41112, 41114, 41123, 41141, 41202, 41213, 41222, 41224, 41233, 41244, 41312, 41321, 41323, 41334, 41404, 41411, 41431, 41444, 42003, 42032, 42102, 42124, 42131, 42142, 42201, 42203, 42234, 42313, 42322, 42333, 42401, 42412, 42421, 43004, 43022, 43042, 43103, 43114, 43132, 43202, 43224, 43303, 43312, 43323, 43334, 43341, 43444, 44001, 44021, 44034, 44043, 44122, 44131, 44144, 44221, 44232, 44304, 44313, 44324, 44414, 44434, 44441, 100022, 100123, 100132, 100134, 100211, 100222, 100231, 100303, 100314, 100332, 100341, 100404, 101001, 101003, 101012, 101014, 101041, 101144, 101201, 101212, 101223, 101234, 101243, 101304, 101311, 101333, 101342, 101414, 101421, 101441, 101443, 102024, 102031, 102112, 102123, 102213, 102244, 102312, 102321, 102323, 102332, 102334, 102431, 102444, 103021, 103032, 103102, 103104, 103113, 103124, 103131, 103142, 103212, 103214, 103241, 103311, 103313, 103333, ...|
|6|11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, 1011, 1015, 1021, 1025, 1035, 1041, 1055, 1105, 1115, 1125, 1131, 1141, 1145, 1151, 1205, 1231, 1235, 1241, 1245, 1311, 1321, 1335, 1341, 1345, 1355, 1411, 1421, 1431, 1435, 1445, 1501, 1505, 1521, 1535, 1541, 1555, 2001, 2011, 2015, 2025, 2041, 2045, 2051, 2055, 2115, 2131, 2135, 2151, 2155, 2205, 2225, 2231, 2301, 2311, 2325, 2335, 2345, 2351, 2401, 2415, 2425, 2435, 2441, 2451, 2501, 2505, 2511, 2531, 2545, 2551, 2555, 3005, 3015, 3021, 3041, 3045, 3055, 3111, 3125, 3141, 3155, 3211, 3221, 3231, 3235, 3251, 3301, 3305, 3321, 3325, 3351, 3405, 3425, 3431, 3445, 3451, 3455, 3501, 3515, 3541, 3545, 3551, 3555, 4021, 4025, 4031, 4035, 4111, 4115, 4131, 4145, 4201, 4205, 4215, 4225, 4251, 4255, 4305, 4315, 4331, 4341, 4401, 4405, 4415, 4421, 4435, 4441, 4451, 4505, 4511, 4525, 4531, 4541, 5011, 5015, 5021, 5025, 5035, 5045, 5101, 5111, 5121, 5155, 5201, 5215, 5231, 5245, 5255, 5305, 5321, 5341, 5345, 5355, 5405, 5411, 5421, 5441, 5455, 5525, 5531, 5535, 5545, 5551, 10001, 10005, 10011, 10015, 10035, 10041, 10051, 10145, 10155, 10205, 10221, 10251, 10305, 10331, 10335, 10341, 10345, 10355, 10411, 10415, 10421, 10431, 10451, 10505, 10511, 10515, 10521, 10525, 10535, 10555, 11015, 11031, 11051, 11101, 11105, 11115, 11131, 11135, 11151, 11155, 11221, 11225, 11235, 11241, 11245, 11255, 11301, 11311, 11325, 11401, 11411, 11415, 11421, 11501, 11505, 11511, 11525, 11545, 11551, 12005, 12021, 12031, 12041, 12051, 12121, 12131, 12135, 12141, 12201, 12215, 12235, 12251, 12315, 12341, 12351, 12355, 12401, 12405, 12411, 12425, 12445, 12455, 12505, 12535, 12541, 13005, 13011, 13045, 13055, 13111, 13121, 13125, 13131, 13135, 13151, 13201, 13215, 13221, 13235, 13301, 13315, 13325, 13345, 13351, 13355, 13401, 13415, 13435, 13441, 13505, 13511, 13521, 13525, 13531, 13545, 14001, 14031, 14111, 14115, 14125, 14141, 14205, 14211, 14215, 14231, 14255, 14301, 14305, 14321, 14331, 14341, 14345, 14405, 14411, 14445, 14455, 14501, 14511, 14515, 14525, 14551, 15001, 15005, 15011, 15021, 15025, 15035, 15055, 15105, 15115, 15141, 15145, 15155, 15215, 15231, 15241, 15245, 15331, 15401, 15415, 15431, 15435, 15445, 15451, 15501, 15535, 15555, 20001, 20025, 20041, 20045, 20105, 20131, 20145, 20151, 20155, 20211, 20221, 20231, 20235, 20241, 20245, 20255, 20311, 20315, 20321, 20331, 20345, 20351, 20405, 20421, 20425, 20451, 20505, 20525, 20531, 20541, 20545, 20551, 21015, 21041, 21045, 21055, 21111, 21121, 21125, 21155, 21211, 21225, 21235, 21245, 21301, 21315, 21335, 21401, 21405, 21415, 21425, 21431, 21515, 21521, 21535, 21551, 21555, 22021, 22025, 22041, 22101, 22111, 22131, 22135, 22145, 22221, 22235, 22241, 22305, 22351, 22355, 22401, 22421, 22431, 22435, 22455, 22505, 22521, 22525, 22541, 23015, 23021, 23025, 23031, 23051, 23135, 23141, 23151, 23201, 23211, 23215, 23225, 23231, 23251, 23255, 23315, 23321, 23335, 23341, 23405, 23411, 23435, 23445, 23521, 23545, 24001, 24005, 24011, 24015, 24021, 24055, 24111, 24131, 24141, 24155, 24201, 24205, 24215, 24221, 24231, 24245, 24251, 24311, 24325, 24331, 24345, ...|
|7|14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 104, 113, 115, 124, 131, 133, 142, 146, 155, 166, 203, 205, 212, 214, 221, 241, 245, 254, 256, 302, 304, 313, 322, 326, 335, 344, 346, 362, 364, 401, 403, 421, 436, 443, 445, 452, 461, 463, 506, 515, 524, 533, 535, 544, 551, 553, 566, 616, 623, 625, 632, 652, 661, 1004, 1006, 1013, 1022, 1033, 1042, 1051, 1055, 1064, 1105, 1112, 1123, 1136, 1141, 1154, 1156, 1165, 1202, 1211, 1222, 1226, 1231, 1235, 1253, 1264, 1301, 1312, 1316, 1325, 1343, 1345, 1402, 1411, 1424, 1433, 1442, 1444, 1453, 1466, 1505, 1514, 1516, 1525, 1534, 1541, 1543, 1561, 1604, 1606, 1613, 1622, 1631, 1633, 1651, 1655, 1664, 2005, 2021, 2032, 2045, 2056, 2065, 2104, 2111, 2122, 2131, 2135, 2146, 2153, 2203, 2216, 2234, 2236, 2252, 2254, 2261, 2263, 2306, 2326, 2333, 2335, 2342, 2362, 2366, 2401, 2405, 2434, 2441, 2452, 2465, 2506, 2513, 2522, 2531, 2551, 2555, 2564, 2603, 2614, 2623, 2641, 2645, 2654, 2656, 3002, 3004, 3013, 3026, 3031, 3044, 3046, 3055, 3112, 3116, 3121, 3125, 3134, 3143, 3154, 3163, 3202, 3233, 3235, 3251, 3262, 3305, 3314, 3323, 3334, 3352, 3356, 3365, 3404, 3406, 3415, 3433, 3446, 3503, 3505, 3512, 3521, 3523, 3532, 3536, 3541, 3545, 3563, 3565, 3604, 3653, 3662, 4001, 4012, 4036, 4052, 4102, 4106, 4111, 4115, 4124, 4135, 4142, 4144, 4153, 4201, 4214, 4216, 4223, 4225, 4232, 4241, 4256, 4304, 4315, 4333, 4342, 4346, 4355, 4366, 4403, 4414, 4421, 4441, 4445, 4454, 4456, 4463, 4502, 4504, 4513, 4526, 4555, 4564, 4601, 4603, 4636, 4643, 4645, 4661, 5006, 5011, 5024, 5035, 5044, 5053, 5062, 5116, 5125, 5132, 5134, 5152, 5165, 5213, 5224, 5246, 5266, 5305, 5312, 5314, 5321, 5323, 5336, 5354, 5363, 5402, 5426, 5431, 5453, 5455, 5516, 5525, 5536, 5545, 5552, 5554, 5561, 5602, 5611, 5624, 5626, 5642, 5662, 6005, 6014, 6032, 6034, 6041, 6043, 6056, 6104, 6106, 6131, 6133, 6142, 6146, 6151, 6164, 6205, 6232, 6265, 6302, 6311, 6322, 6344, 6346, 6353, 6364, 6416, 6421, 6425, 6436, 6445, 6454, 6461, 6506, 6511, 6542, 6551, 6553, 6562, 6566, 6605, 6625, 6634, 6641, 6643, 6652, 6656, 6665, 10013, 10022, 10031, 10051, 10055, 10064, 10112, 10123, 10132, 10136, 10204, 10231, 10244, 10255, 10262, 10301, 10303, 10312, 10343, 10361, 10363, 10415, 10426, 10433, 10451, 10501, 10514, 10516, 10523, 10534, 10543, 10552, 10556, 10561, 10565, 10604, 10615, 10622, 10624, 10633, 10646, 10651, 10664, 11005, 11012, 11032, 11045, 11063, 11065, 11104, 11111, 11113, 11135, 11155, 11162, 11201, 11212, 11221, 11225, 11252, 11263, 11306, 11315, 11324, 11335, 11351, 11366, 11416, 11423, 11432, 11441, 11443, 11513, 11515, 11531, 11542, 11546, 11566, 11603, 11614, 11632, 11641, 11656, 11663, 12002, 12031, 12044, 12046, 12101, 12136, 12143, 12145, 12163, 12202, 12206, 12224, 12233, 12244, 12251, 12262, 12323, 12325, 12332, 12334, 12352, 12422, 12424, 12433, 12442, 12451, 12455, 12464, 12466, 12514, 12521, 12536, 12541, 12554, 12556, 12611, 12613, 12635, 12644, 13003, 13025, 13036, 13043, 13045, 13052, 13054, 13115, 13126, 13144, 13153, 13166, 13201, 13205, 13214, 13216, 13225, 13241, 13243, 13261, 13304, 13306, 13322, 13342, ...|
|8|13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 103, 107, 111, 117, 123, 131, 141, 145, 147, 153, 155, 161, 177, 203, 211, 213, 225, 227, 235, 243, 247, 255, 263, 265, 277, 301, 305, 307, 323, 337, 343, 345, 351, 357, 361, 373, 401, 407, 415, 417, 425, 431, 433, 445, 463, 467, 471, 475, 513, 521, 533, 535, 541, 547, 557, 565, 573, 577, 605, 615, 621, 631, 643, 645, 657, 661, 667, 673, 701, 711, 715, 717, 723, 737, 747, 753, 763, 767, 775, 1011, 1013, 1035, 1043, 1055, 1063, 1071, 1073, 1101, 1113, 1121, 1127, 1131, 1137, 1145, 1151, 1153, 1167, 1201, 1203, 1207, 1215, 1223, 1225, 1241, 1245, 1253, 1263, 1275, 1305, 1317, 1327, 1335, 1343, 1347, 1357, 1365, 1371, 1401, 1405, 1423, 1435, 1451, 1453, 1465, 1467, 1473, 1475, 1507, 1525, 1531, 1533, 1537, 1555, 1561, 1563, 1567, 1613, 1617, 1627, 1641, 1651, 1655, 1663, 1671, 1707, 1713, 1721, 1727, 1737, 1745, 1761, 1765, 1773, 1775, 2007, 2011, 2017, 2031, 2033, 2045, 2047, 2055, 2077, 2103, 2105, 2111, 2117, 2125, 2135, 2143, 2151, 2177, 2201, 2213, 2223, 2235, 2243, 2251, 2261, 2275, 2301, 2307, 2315, 2317, 2325, 2341, 2353, 2375, 2377, 2403, 2411, 2413, 2421, 2425, 2427, 2433, 2447, 2451, 2457, 2521, 2527, 2535, 2545, 2567, 2601, 2617, 2623, 2625, 2631, 2637, 2647, 2653, 2655, 2663, 2677, 2711, 2713, 2717, 2721, 2725, 2733, 2747, 2763, 2773, 3007, 3015, 3021, 3027, 3037, 3043, 3053, 3057, 3075, 3101, 3107, 3111, 3115, 3123, 3125, 3133, 3145, 3171, 3177, 3203, 3205, 3235, 3241, 3243, 3255, 3271, 3273, 3305, 3315, 3323, 3331, 3337, 3361, 3367, 3373, 3375, 3411, 3423, 3437, 3447, 3467, 3505, 3513, 3517, 3521, 3525, 3527, 3541, 3555, 3563, 3571, 3613, 3615, 3635, 3637, 3665, 3673, 3703, 3711, 3715, 3717, 3723, 3733, 3741, 3753, 3755, 3767, 4005, 4017, 4025, 4041, 4043, 4047, 4051, 4063, 4077, 4101, 4121, 4123, 4131, 4135, 4137, 4151, 4161, 4203, 4233, 4237, 4245, 4255, 4275, 4277, 4303, 4313, 4333, 4335, 4341, 4351, 4357, 4365, 4371, 4405, 4407, 4435, 4443, 4445, 4453, 4457, 4465, 4503, 4511, 4515, 4517, 4525, 4531, 4537, 4553, 4561, 4567, 4605, 4611, 4617, 4633, 4643, 4651, 4655, 4707, 4731, 4743, 4753, 4757, 4765, 4767, 4775, 5023, 5037, 5041, 5061, 5071, 5075, 5111, 5127, 5141, 5143, 5147, 5157, 5165, 5173, 5177, 5201, 5205, 5213, 5223, 5227, 5231, 5237, 5251, 5253, 5265, 5275, 5301, 5317, 5331, 5345, 5347, 5355, 5361, 5363, 5403, 5421, 5425, 5433, 5443, 5451, 5455, 5477, 5507, 5521, 5527, 5535, 5545, 5557, 5573, 5611, 5615, 5623, 5631, 5633, 5667, 5671, 5703, 5713, 5717, 5735, 5741, 5751, 5765, 5773, 6007, 6013, 6021, 6045, 6057, 6061, 6101, 6133, 6137, 6141, 6155, 6163, 6167, 6203, 6211, 6221, 6225, 6235, 6263, 6265, 6271, 6273, 6307, 6343, 6345, 6353, 6361, 6367, 6373, 6401, 6403, 6417, 6423, 6437, 6441, 6453, 6455, 6475, 6477, 6517, 6525, 6551, 6571, 6601, 6605, 6607, 6613, 6615, 6643, 6653, 6667, 6675, 6707, 6711, 6715, 6723, 6725, 6733, 6745, 6747, 6763, 6775, 6777, 7011, 7027, ...|
|9|12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 102, 108, 117, 122, 124, 128, 131, 135, 151, 155, 162, 164, 175, 177, 184, 201, 205, 212, 218, 221, 232, 234, 238, 241, 254, 267, 272, 274, 278, 285, 287, 308, 315, 322, 328, 331, 337, 342, 344, 355, 371, 375, 377, 382, 407, 414, 425, 427, 432, 438, 447, 454, 461, 465, 472, 481, 485, 504, 515, 517, 528, 531, 537, 542, 548, 557, 562, 564, 568, 582, 601, 605, 614, 618, 625, 638, 641, 661, 667, 678, 685, 702, 704, 711, 722, 728, 735, 737, 744, 751, 755, 757, 771, 782, 784, 788, 805, 812, 814, 827, 832, 838, 847, 858, 867, 878, 887, 1004, 1011, 1015, 1024, 1031, 1035, 1044, 1048, 1064, 1075, 1088, 1101, 1112, 1114, 1118, 1121, 1132, 1147, 1152, 1154, 1158, 1174, 1178, 1181, 1185, 1217, 1222, 1231, 1242, 1251, 1255, 1262, 1268, 1284, 1288, 1305, 1312, 1321, 1327, 1341, 1345, 1352, 1354, 1365, 1367, 1374, 1385, 1387, 1408, 1411, 1417, 1437, 1442, 1444, 1448, 1455, 1462, 1471, 1477, 1484, 1518, 1521, 1532, 1541, 1552, 1558, 1565, 1574, 1587, 1602, 1608, 1615, 1617, 1624, 1637, 1648, 1668, 1671, 1675, 1682, 1684, 1701, 1705, 1707, 1712, 1725, 1727, 1734, 1772, 1778, 1785, 1804, 1824, 1835, 1851, 1855, 1857, 1862, 1868, 1877, 1882, 1884, 2001, 2014, 2025, 2027, 2032, 2034, 2038, 2045, 2058, 2072, 2081, 2104, 2111, 2115, 2122, 2131, 2135, 2144, 2148, 2164, 2168, 2175, 2177, 2182, 2188, 2201, 2207, 2218, 2241, 2247, 2252, 2254, 2281, 2285, 2287, 2308, 2322, 2324, 2335, 2344, 2351, 2357, 2364, 2384, 2401, 2405, 2407, 2421, 2432, 2445, 2454, 2472, 2487, 2504, 2508, 2511, 2515, 2517, 2528, 2542, 2548, 2555, 2575, 2577, 2605, 2607, 2632, 2638, 2647, 2654, 2658, 2661, 2665, 2674, 2681, 2702, 2704, 2715, 2731, 2742, 2748, 2762, 2764, 2768, 2771, 2782, 2805, 2807, 2825, 2827, 2834, 2838, 2841, 2852, 2861, 2881, 3017, 3022, 3028, 3037, 3055, 3057, 3062, 3071, 3088, 3101, 3105, 3114, 3121, 3127, 3132, 3145, 3147, 3172, 3178, 3181, 3187, 3202, 3208, 3224, 3231, 3235, 3237, 3244, 3248, 3255, 3268, 3275, 3282, 3307, 3312, 3318, 3332, 3341, 3347, 3352, 3381, 3411, 3422, 3431, 3435, 3442, 3444, 3451, 3475, 3488, 3501, 3518, 3527, 3532, 3545, 3561, 3572, 3574, 3578, 3587, 3604, 3611, 3615, 3617, 3622, 3628, 3637, 3642, 3644, 3651, 3662, 3664, 3675, 3684, 3688, 3714, 3725, 3738, 3741, 3747, 3752, 3754, 3772, 3787, 3802, 3808, 3817, 3824, 3828, 3848, 3857, 3868, 3875, 3882, 4001, 4012, 4025, 4041, 4045, 4052, 4058, 4061, 4102, 4104, 4115, 4124, 4128, 4144, 4148, 4157, 4171, 4177, 4201, 4205, 4212, 4234, 4245, 4247, 4265, 4304, 4308, 4311, 4324, 4331, 4335, 4348, 4355, 4364, 4368, 4377, 4412, 4414, 4418, 4421, 4434, 4465, 4467, 4474, 4481, 4487, 4502, 4508, 4511, 4524, 4528, 4542, 4544, 4555, 4557, 4575, 4577, 4605, 4612, 4634, 4652, 4661, 4665, 4667, 4672, 4674, 4708, 4717, 4731, 4737, 4748, 4751, 4755, 4762, 4764, 4771, 4782, 4784, 4807, 4818, 4821, 4832, 4847, ...|
|10|11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, ...|
|11|12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 106, 10A, 115, 117, 126, 128, 133, 139, 142, 148, 153, 155, 164, 166, 16A, 171, 182, 193, 197, 199, 1A2, 1A8, 1AA, 209, 214, 21A, 225, 227, 232, 236, 238, 247, 25A, 263, 265, 269, 281, 287, 296, 298, 2A1, 2A7, 304, 30A, 315, 319, 324, 331, 335, 342, 351, 353, 362, 364, 36A, 373, 379, 386, 38A, 391, 395, 3A6, 403, 407, 414, 418, 423, 434, 436, 452, 458, 467, 472, 478, 47A, 485, 494, 49A, 4A5, 4A7, 502, 508, 511, 513, 524, 533, 535, 539, 544, 54A, 551, 562, 566, 571, 579, 588, 595, 5A4, 601, 607, 612, 616, 623, 629, 632, 63A, 643, 656, 665, 676, 678, 687, 689, 692, 694, 6A3, 706, 70A, 711, 715, 728, 731, 733, 737, 755, 759, 766, 775, 782, 786, 791, 797, 7AA, 803, 809, 814, 821, 827, 838, 841, 847, 849, 858, 85A, 865, 874, 876, 885, 887, 892, 8A9, 902, 904, 908, 913, 919, 926, 931, 937, 957, 959, 968, 975, 984, 98A, 995, 9A2, A03, A07, A12, A18, A1A, A25, A36, A45, A61, A63, A67, A72, A74, A7A, A83, A85, A89, A9A, AA1, AA7, 1028, 1033, 1039, 1046, 1062, 1071, 1084, 1088, 108A, 1093, 1099, 10A6, 10AA, 1101, 1107, 1118, 1127, 1129, 1132, 1134, 1138, 1143, 1154, 1165, 1172, 1183, 1189, 1192, 1198, 11A5, 11A9, 1206, 120A, 1222, 1226, 1231, 1233, 1237, 1242, 1244, 124A, 1259, 1277, 1282, 1286, 1288, 12AA, 1303, 1305, 1314, 1325, 1327, 1336, 1343, 1349, 1354, 135A, 1376, 1381, 1385, 1387, 1398, 13A7, 1408, 1415, 142A, 1442, 1448, 1451, 1453, 1457, 1459, 1468, 1479, 1484, 148A, 14A6, 14A8, 1512, 1514, 1534, 153A, 1547, 1552, 1556, 1558, 1561, 1569, 1574, 1583, 1585, 1594, 15A7, 1606, 1611, 1622, 1624, 1628, 162A, 1639, 164A, 1651, 1666, 1668, 1673, 1677, 1679, 1688, 1695, 1701, 1723, 1727, 1732, 173A, 1754, 1756, 175A, 1767, 1781, 1783, 1787, 1794, 179A, 17A5, 17A9, 180A, 1811, 1831, 1837, 1839, 1844, 1848, 1853, 1866, 1871, 1875, 1877, 1882, 1886, 1891, 18A2, 18A8, 1903, 1916, 191A, 1925, 1936, 1943, 1949, 1952, 1976, 1992, 19A1, 19A9, 1A02, 1A08, 1A0A, 1A15, 1A35, 1A46, 1A48, 1A62, 1A6A, 1A73, 1A84, 1A97, 1AA6, 1AA8, 2001, 2009, 2014, 201A, 2023, 2025, 2029, 2034, 2041, 2045, 2047, 2052, 2061, 2063, 2072, 207A, 2083, 2096, 20A5, 2106, 2108, 2113, 2117, 2119, 2133, 2146, 214A, 2155, 2162, 2168, 2171, 2188, 2195, 21A4, 21AA, 2205, 2212, 2221, 2232, 2245, 2249, 2254, 225A, 2261, 2287, 2289, 2298, 22A5, 22A9, 2311, 2315, 2322, 2333, 2339, 234A, 2353, 2359, 2377, 2386, 2388, 23A2, 2416, 241A, 2421, 2432, 2438, 2441, 2452, 2458, 2465, 2469, 2476, 2496, 2498, 24A1, 24A3, 2504, 252A, 2531, 2537, 2542, 2548, 2551, 2557, 2559, 256A, 2573, 2584, 2586, 2595, 2597, 2601, 2603, 2618, 2623, 2641, 2656, 2663, 2667, 2669, 2672, 2674, 2694, 26A1, 2702, 2708, 2717, 2719, 2722, 2728, 272A, 2735, 2744, 2746, 2757, 2766, 2768, 2777, 278A, 2795, ...|
|12|11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 105, 107, 111, 117, 11B, 125, 12B, 131, 13B, 141, 145, 147, 157, 167, 16B, 171, 175, 17B, 181, 18B, 195, 19B, 1A5, 1A7, 1B1, 1B5, 1B7, 205, 217, 21B, 221, 225, 237, 241, 24B, 251, 255, 25B, 267, 271, 277, 27B, 285, 291, 295, 2A1, 2AB, 2B1, 2BB, 301, 307, 30B, 315, 321, 325, 327, 32B, 33B, 347, 34B, 357, 35B, 365, 375, 377, 391, 397, 3A5, 3AB, 3B5, 3B7, 401, 40B, 415, 41B, 421, 427, 431, 435, 437, 447, 455, 457, 45B, 465, 46B, 471, 481, 485, 48B, 497, 4A5, 4B1, 4BB, 507, 511, 517, 51B, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58B, 591, 59B, 5B1, 5B5, 5B7, 5BB, 611, 615, 617, 61B, 637, 63B, 647, 655, 661, 665, 66B, 675, 687, 68B, 695, 69B, 6A7, 6B1, 701, 705, 70B, 711, 71B, 721, 727, 735, 737, 745, 747, 751, 767, 76B, 771, 775, 77B, 785, 791, 797, 7A1, 7BB, 801, 80B, 817, 825, 82B, 835, 841, 851, 855, 85B, 865, 867, 871, 881, 88B, 8A5, 8A7, 8AB, 8B5, 8B7, 901, 905, 907, 90B, 91B, 921, 927, 955, 95B, 965, 971, 987, 995, 9A7, 9AB, 9B1, 9B5, 9BB, A07, A0B, A11, A17, A27, A35, A37, A3B, A41, A45, A4B, A5B, A6B, A77, A87, A91, A95, A9B, AA7, AAB, AB7, ABB, B11, B15, B1B, B21, B25, B2B, B31, B37, B45, B61, B67, B6B, B71, B91, B95, B97, BA5, BB5, BB7, 1005, 1011, 1017, 1021, 1027, 1041, 1047, 104B, 1051, 1061, 106B, 107B, 1087, 109B, 10B1, 10B7, 10BB, 1101, 1105, 1107, 1115, 1125, 112B, 1135, 114B, 1151, 1165, 1167, 1185, 118B, 1197, 11A1, 11A5, 11A7, 11AB, 11B7, 1201, 120B, 1211, 121B, 1231, 123B, 1245, 1255, 1257, 125B, 1261, 126B, 127B, 1281, 1295, 1297, 12A1, 12A5, 12A7, 12B5, 1301, 1317, 1337, 133B, 1345, 1351, 1365, 1367, 136B, 1377, 138B, 1391, 1395, 13A1, 13A7, 13B1, 13B5, 1405, 1407, 1425, 142B, 1431, 1437, 143B, 1445, 1457, 1461, 1465, 1467, 1471, 1475, 147B, 148B, 1495, 149B, 14B1, 14B5, 14BB, 150B, 1517, 1521, 1525, 1547, 1561, 156B, 1577, 157B, 1585, 1587, 1591, 15AB, 15BB, 1601, 1615, 1621, 1625, 1635, 1647, 1655, 1657, 165B, 1667, 1671, 1677, 167B, 1681, 1685, 168B, 1697, 169B, 16A1, 16A7, 16B5, 16B7, 1705, 1711, 1715, 1727, 1735, 1745, 1747, 1751, 1755, 1757, 176B, 1781, 1785, 178B, 1797, 17A1, 17A5, 17BB, 1807, 1815, 181B, 1825, 1831, 183B, 184B, 1861, 1865, 186B, 1875, 1877, 189B, 18A1, 18AB, 18B7, 18BB, 1911, 1915, 1921, 1931, 1937, 1947, 194B, 1955, 1971, 197B, 1981, 1995, 19B7, 19BB, 1A01, 1A11, 1A17, 1A1B, 1A2B, 1A35, 1A41, 1A45, 1A51, 1A6B, 1A71, 1A75, 1A77, 1A87, 1AAB, 1AB1, 1AB7, 1B01, 1B07, 1B0B, 1B15, 1B17, 1B27, 1B2B, 1B3B, 1B41, 1B4B, 1B51, 1B65, 1B67, 1B7B, 1B85, 1BA1, 1BB5, 2001, 2005, 2007, 200B, 2011, 202B, 2037, 2047, 2051, 205B, 2061, 2065, 206B, 2071, 2077, 2085, 2087, 2097, 20A5, 20A7, 20B5, 2107, 2111, ...|
|13|14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 104, 10A, 10C, 119, 11B, 122, 124, 133, 142, 146, 148, 14C, 155, 157, 164, 16A, 173, 179, 17B, 184, 188, 18A, 197, 1A8, 1AC, 1B1, 1B5, 1C6, 1CC, 209, 20B, 212, 218, 223, 229, 232, 236, 23C, 247, 24B, 256, 263, 265, 272, 274, 27A, 281, 287, 292, 296, 298, 29C, 2AB, 2B6, 2BA, 2C5, 2C9, 302, 311, 313, 328, 331, 33B, 344, 34A, 34C, 355, 362, 368, 371, 373, 379, 382, 386, 388, 397, 3A4, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 401, 407, 412, 41C, 427, 434, 43C, 445, 44B, 452, 45A, 463, 467, 472, 476, 487, 494, 4A3, 4A5, 4B2, 4B4, 4B8, 4BA, 4C7, 508, 50C, 511, 515, 526, 52A, 52C, 533, 54A, 551, 559, 566, 571, 575, 57B, 584, 595, 599, 5A2, 5A8, 5B3, 5B9, 5C8, 5CC, 605, 607, 614, 616, 61C, 629, 62B, 638, 63A, 643, 658, 65C, 661, 665, 66B, 674, 67C, 685, 68B, 6A7, 6A9, 6B6, 6C1, 6CB, 704, 70A, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 78A, 791, 793, 797, 7A6, 7A8, 7B1, 809, 812, 818, 823, 838, 845, 856, 85A, 85C, 863, 869, 874, 878, 87A, 883, 892, 89C, 8A1, 8A5, 8A7, 8AB, 8B4, 8C3, 902, 90A, 919, 922, 926, 92C, 937, 93B, 946, 94A, 95B, 962, 968, 96A, 971, 977, 979, 982, 98C, 9A6, 9AC, 9B3, 9B5, A03, A07, A09, A16, A25, A27, A34, A3C, A45, A4B, A54, A69, A72, A76, A78, A87, A94, AA3, AAB, AC1, B02, B08, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B6C, B71, B8A, B93, B9B, BA4, BA8, BAA, BB1, BB9, BC2, BCC, C01, C0B, C1C, C29, C32, C41, C43, C47, C49, C56, C65, C67, C7A, C7C, C85, C89, C8B, C98, CA3, CB8, 1006, 100A, 1013, 101B, 1031, 1033, 1037, 1042, 1055, 1057, 105B, 1066, 106C, 1075, 1079, 1088, 108A, 10A6, 10AC, 10B1, 10B7, 10BB, 10C4, 1105, 110B, 1112, 1114, 111A, 1121, 1127, 1136, 113C, 1145, 1156, 115A, 1163, 1172, 117A, 1183, 1187, 11A7, 11BC, 11C9, 1204, 1208, 1211, 1213, 1219, 1235, 1244, 1246, 1259, 1264, 1268, 1277, 1288, 1295, 1297, 129B, 12A6, 12AC, 12B5, 12B9, 12BB, 12C2, 12C8, 1303, 1307, 1309, 1312, 131C, 1321, 132B, 1336, 133A, 134B, 1358, 1367, 1369, 1372, 1376, 1378, 138B, 139C, 13A3, 13A9, 13B4, 13BA, 13C1, 1406, 1411, 141B, 1424, 142A, 1435, 1442, 1451, 1462, 1466, 146C, 1475, 1477, 1499, 149B, 14A8, 14B3, 14B7, 14C8, 14CC, 1507, 1516, 151C, 152B, 1532, 1538, 1552, 155C, 1561, 1574, 1594, 1598, 159A, 15A9, 15B2, 15B6, 15C5, 15CB, 1606, 160A, 1615, 1631, 1633, 1637, 1639, 1648, 166A, 166C, 1675, 167B, 1684, 1688, 1691, 1693, 16A2, 16A6, 16B5, 16B7, 16C4, 16C6, 1709, 170B, 1721, 1727, 1741, 1754, 175C, 1763, 1765, 1769, 176B, 1787, 1792, 17A1, 17A7, 17B4, 17B6, 17BA, 17C3, 17C5, 17CB, 1808, 180A, 1819, 1826, 1828, 1835, 1846, 184C, 1853, ...|
|14|13, 15, 19, 21, 23, 29, 2D, 31, 35, 3B, 43, 45, 4B, 51, 53, 59, 5D, 65, 6D, 73, 75, 79, 7B, 81, 91, 95, 9B, 9D, A9, AB, B3, B9, BD, C5, CB, CD, D9, DB, 101, 103, 111, 11D, 123, 125, 129, 131, 133, 13D, 145, 14B, 153, 155, 15B, 161, 163, 16D, 17D, 183, 185, 189, 199, 1A1, 1AB, 1AD, 1B3, 1B9, 1C3, 1C9, 1D1, 1D5, 1DB, 205, 209, 213, 21D, 221, 22B, 22D, 235, 239, 241, 249, 24D, 251, 255, 263, 26B, 271, 279, 27D, 285, 293, 295, 2A9, 2B1, 2BB, 2C3, 2C9, 2CB, 2D3, 2DD, 305, 30B, 30D, 315, 31B, 321, 323, 331, 33B, 33D, 343, 349, 351, 353, 361, 365, 36B, 375, 381, 389, 395, 39D, 3A5, 3AB, 3B1, 3B9, 3C1, 3C5, 3CD, 3D3, 403, 40D, 41B, 41D, 429, 42B, 431, 433, 43D, 44D, 453, 455, 459, 469, 46D, 471, 475, 48B, 491, 499, 4A5, 4AD, 4B3, 4B9, 4C1, 4D1, 4D5, 4DB, 503, 50B, 513, 521, 525, 52B, 52D, 539, 53B, 543, 54D, 551, 55B, 55D, 565, 579, 57D, 581, 585, 58B, 593, 59B, 5A3, 5A9, 5C3, 5C5, 5D1, 5D9, 605, 60B, 613, 61B, 629, 62D, 635, 63B, 63D, 645, 653, 65D, 673, 675, 679, 681, 683, 689, 68D, 691, 695, 6A3, 6A5, 6AB, 6D3, 6D9, 701, 709, 71D, 729, 739, 73D, 741, 745, 74B, 755, 759, 75B, 763, 771, 77B, 77D, 783, 785, 789, 791, 79D, 7AB, 7B5, 7C3, 7C9, 7CD, 7D5, 7DD, 803, 80B, 811, 821, 825, 82B, 82D, 833, 839, 83B, 843, 84D, 865, 86B, 871, 873, 88D, 893, 895, 8A1, 8AD, 8B1, 8BB, 8C5, 8CB, 8D3, 8D9, 90D, 915, 919, 91B, 929, 935, 943, 94B, 95D, 96D, 975, 979, 97B, 981, 983, 98D, 99B, 9A3, 9A9, 9BD, 9C1, 9D3, 9D5, A0D, A15, A1D, A25, A29, A2B, A31, A39, A41, A4B, A4D, A59, A69, A75, A7B, A89, A8B, A91, A93, A9D, AAB, AAD, AC1, AC3, AC9, ACD, AD1, ADB, B05, B19, B35, B39, B41, B49, B5B, B5D, B63, B6B, B7D, B81, B85, B8D, B95, B9B, BA1, BAD, BB1, BC9, BD1, BD3, BD9, BDD, C05, C15, C1B, C21, C23, C29, C2D, C35, C43, C49, C51, C61, C65, C6B, C79, C83, C89, C8D, CAB, CC1, CCB, CD5, CD9, D01, D03, D09, D23, D31, D33, D45, D4D, D53, D61, D71, D7B, D7D, D83, D8B, D93, D99, D9D, DA1, DA5, DAB, DB5, DB9, DBB, DC3, DCD, DD1, DDB, 1005, 1009, 1019, 1025, 1033, 1035, 103B, 1041, 1043, 1055, 1065, 1069, 1071, 1079, 1081, 1085, 1099, 10A3, 10AD, 10B5, 10BB, 10C5, 10D1, 10DD, 110D, 1113, 1119, 1121, 1123, 1143, 1145, 1151, 1159, 115D, 116D, 1173, 117B, 1189, 1191, 119D, 11A3, 11A9, 11C1, 11CB, 11CD, 1201, 121D, 1223, 1225, 1233, 1239, 123D, 124B, 1253, 125B, 1261, 1269, 1283, 1285, 1289, 128B, 1299, 12B9, 12BB, 12C3, 12C9, 12D1, 12D5, 12DB, 12DD, 130B, 1311, 131D, 1321, 132B, 132D, 1341, 1343, 1355, 135B, 1373, 1385, 138D, 1393, 1395, 1399, 139B, 13B5, 13BD, 13CB, 13D3, 13DD, 1401, 1405, 140B, 140D, 1415, 1421, 1423, 1431, 143B, 143D, 1449, 1459, 1461, 1465, ...|
|15|12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 102, 104, 108, 10E, 111, 11B, 122, 128, 12E, 131, 137, 13B, 13D, 148, 157, 15B, 15D, 162, 171, 177, 182, 184, 188, 18E, 197, 19D, 1A4, 1A8, 1AE, 1B7, 1BB, 1C4, 1CE, 1D1, 1DB, 1DD, 1E4, 1E8, 1EE, 207, 20B, 20D, 212, 21E, 227, 22B, 234, 238, 23E, 24B, 24D, 261, 267, 272, 278, 27E, 281, 287, 292, 298, 29E, 2A1, 2A7, 2AD, 2B2, 2B4, 2C1, 2CB, 2CD, 2D2, 2D8, 2DE, 2E1, 2ED, 302, 308, 311, 31B, 324, 32E, 337, 33D, 344, 348, 351, 357, 35B, 364, 368, 377, 382, 38E, 391, 39B, 39D, 3A2, 3A4, 3AE, 3BD, 3C2, 3C4, 3C8, 3D7, 3DB, 3DD, 3E2, 407, 40B, 414, 41E, 427, 42B, 432, 438, 447, 44B, 452, 458, 461, 467, 474, 478, 47E, 481, 48B, 48D, 494, 49E, 4A1, 4AB, 4AD, 4B4, 4C7, 4CB, 4CD, 4D2, 4D8, 4DE, 4E7, 4ED, 504, 51B, 51D, 528, 531, 53B, 542, 548, 551, 55D, 562, 568, 56E, 571, 577, 584, 58E, 5A2, 5A4, 5A8, 5AE, 5B1, 5B7, 5BB, 5BD, 5C2, 5CE, 5D1, 5D7, 60B, 612, 618, 621, 634, 63E, 64D, 652, 654, 658, 65E, 667, 66B, 66D, 674, 681, 68B, 68D, 692, 694, 698, 69E, 6AB, 6B8, 6C1, 6CD, 6D4, 6D8, 6DE, 6E7, 6EB, 704, 708, 717, 71B, 722, 724, 728, 72E, 731, 737, 742, 757, 75D, 762, 764, 77D, 782, 784, 78E, 79B, 79D, 7A8, 7B1, 7B7, 7BD, 7C4, 7D7, 7DD, 7E2, 7E4, 801, 80B, 818, 821, 832, 841, 847, 84B, 84D, 852, 854, 85E, 86B, 872, 878, 88B, 88D, 89E, 8A1, 8B8, 8BE, 8C7, 8CD, 8D2, 8D4, 8D8, 8E1, 8E7, 902, 904, 90E, 91D, 928, 92E, 93B, 93D, 942, 944, 94E, 95B, 95D, 96E, 971, 977, 97B, 97D, 988, 991, 9A4, 9BD, 9C2, 9C8, 9D1, 9E2, 9E4, 9E8, A01, A12, A14, A18, A21, A27, A2D, A32, A3E, A41, A58, A5E, A61, A67, A6B, A72, A81, A87, A8B, A8D, A94, A98, A9E, AAB, AB2, AB8, AC7, ACB, AD2, ADE, AE7, AED, B02, B1D, B31, B3B, B44, B48, B4E, B51, B57, B6E, B7B, B7D, B8E, B97, B9B, BA8, BB7, BC2, BC4, BC8, BD1, BD7, BDD, BE2, BE4, BE8, BEE, C07, C0B, C0D, C14, C1E, C21, C2B, C34, C38, C47, C52, C5E, C61, C67, C6B, C6D, C7E, C8D, C92, C98, CA1, CA7, CAB, CBE, CC7, CD2, CD8, CDE, CE7, D02, D0E, D1D, D22, D28, D2E, D31, D4E, D51, D5B, D64, D68, D77, D7B, D84, D91, D97, DA4, DA8, DAE, DC4, DCE, DD1, DE2, E0D, E12, E14, E21, E27, E2B, E38, E3E, E47, E4B, E54, E6B, E6D, E72, E74, E81, E9E, EA1, EA7, EAD, EB4, EB8, EBE, EC1, ECD, ED2, EDE, EE1, EEB, EED, 100E, 1011, 1022, 1028, 103D, 104E, 1057, 105B, 105D, 1062, 1064, 107B, 1084, 1091, 1097, 10A2, 10A4, 10A8, 10AE, 10B1, 10B7, 10C2, 10C4, 10D1, 10DB, 10DD, 10E8, 1107, 110D, 1112, ...|
|16|11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 101, 107, 10D, 10F, 115, 119, 11B, 125, 133, 137, 139, 13D, 14B, 151, 15B, 15D, 161, 167, 16F, 175, 17B, 17F, 185, 18D, 191, 199, 1A3, 1A5, 1AF, 1B1, 1B7, 1BB, 1C1, 1C9, 1CD, 1CF, 1D3, 1DF, 1E7, 1EB, 1F3, 1F7, 1FD, 209, 20B, 21D, 223, 22D, 233, 239, 23B, 241, 24B, 251, 257, 259, 25F, 265, 269, 26B, 277, 281, 283, 287, 28D, 293, 295, 2A1, 2A5, 2AB, 2B3, 2BD, 2C5, 2CF, 2D7, 2DD, 2E3, 2E7, 2EF, 2F5, 2F9, 301, 305, 313, 31D, 329, 32B, 335, 337, 33B, 33D, 347, 355, 359, 35B, 35F, 36D, 371, 373, 377, 38B, 38F, 397, 3A1, 3A9, 3AD, 3B3, 3B9, 3C7, 3CB, 3D1, 3D7, 3DF, 3E5, 3F1, 3F5, 3FB, 3FD, 407, 409, 40F, 419, 41B, 425, 427, 42D, 43F, 443, 445, 449, 44F, 455, 45D, 463, 469, 47F, 481, 48B, 493, 49D, 4A3, 4A9, 4B1, 4BD, 4C1, 4C7, 4CD, 4CF, 4D5, 4E1, 4EB, 4FD, 4FF, 503, 509, 50B, 511, 515, 517, 51B, 527, 529, 52F, 551, 557, 55D, 565, 577, 581, 58F, 593, 595, 599, 59F, 5A7, 5AB, 5AD, 5B3, 5BF, 5C9, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 5F3, 5FB, 607, 60D, 611, 617, 61F, 623, 62B, 62F, 63D, 641, 647, 649, 64D, 653, 655, 65B, 665, 679, 67F, 683, 685, 69D, 6A1, 6A3, 6AD, 6B9, 6BB, 6C5, 6CD, 6D3, 6D9, 6DF, 6F1, 6F7, 6FB, 6FD, 709, 713, 71F, 727, 737, 745, 74B, 74F, 751, 755, 757, 761, 76D, 773, 779, 78B, 78D, 79D, 79F, 7B5, 7BB, 7C3, 7C9, 7CD, 7CF, 7D3, 7DB, 7E1, 7EB, 7ED, 7F7, 805, 80F, 815, 821, 823, 827, 829, 833, 83F, 841, 851, 853, 859, 85D, 85F, 869, 871, 883, 89B, 89F, 8A5, 8AD, 8BD, 8BF, 8C3, 8CB, 8DB, 8DD, 8E1, 8E9, 8EF, 8F5, 8F9, 905, 907, 91D, 923, 925, 92B, 92F, 935, 943, 949, 94D, 94F, 955, 959, 95F, 96B, 971, 977, 985, 989, 98F, 99B, 9A3, 9A9, 9AD, 9C7, 9D9, 9E3, 9EB, 9EF, 9F5, 9F7, 9FD, A13, A1F, A21, A31, A39, A3D, A49, A57, A61, A63, A67, A6F, A75, A7B, A7F, A81, A85, A8B, A93, A97, A99, A9F, AA9, AAB, AB5, ABD, AC1, ACF, AD9, AE5, AE7, AED, AF1, AF3, B03, B11, B15, B1B, B23, B29, B2D, B3F, B47, B51, B57, B5D, B65, B6F, B7B, B89, B8D, B93, B99, B9B, BB7, BB9, BC3, BCB, BCF, BDD, BE1, BE9, BF5, BFB, C07, C0B, C11, C25, C2F, C31, C41, C5B, C5F, C61, C6D, C73, C77, C83, C89, C91, C95, C9D, CB3, CB5, CB9, CBB, CC7, CE3, CE5, CEB, CF1, CF7, CFB, D01, D03, D0F, D13, D1F, D21, D2B, D2D, D3D, D3F, D4F, D55, D69, D79, D81, D85, D87, D8B, D8D, DA3, DAB, DB7, DBD, DC7, DC9, DCD, DD3, DD5, DDB, DE5, DE7, DF3, DFD, DFF, E09, E17, E1D, E21, ...|
|17|12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 128, 12E, 137, 139, 13D, 142, 14A, 14G, 155, 159, 15F, 166, 16A, 171, 17B, 17D, 186, 188, 18E, 191, 197, 19F, 1A2, 1A4, 1A8, 1B3, 1BB, 1BF, 1C6, 1CA, 1CG, 1DB, 1DD, 1EE, 1F3, 1FD, 1G2, 1G8, 1GA, 1GG, 209, 20F, 214, 216, 21C, 221, 225, 227, 232, 23C, 23E, 241, 247, 24D, 24F, 25A, 25E, 263, 26B, 274, 27C, 285, 28D, 292, 298, 29C, 2A3, 2A9, 2AD, 2B4, 2B8, 2C5, 2CF, 2DA, 2DC, 2E5, 2E7, 2EB, 2ED, 2F6, 2G3, 2G7, 2G9, 2GD, 30A, 30E, 30G, 313, 326, 32A, 331, 33B, 342, 346, 34C, 351, 35F, 362, 368, 36E, 375, 37B, 386, 38A, 38G, 391, 39B, 39D, 3A2, 3AC, 3AE, 3B7, 3B9, 3BF, 3CG, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 3GC, 3GE, 407, 40F, 418, 41E, 423, 42B, 436, 43A, 43G, 445, 447, 44D, 458, 461, 472, 474, 478, 47E, 47G, 485, 489, 48B, 48F, 49A, 49C, 4A1, 4C1, 4C7, 4CD, 4D4, 4E5, 4EF, 4FC, 4FG, 4G1, 4G5, 4GB, 502, 506, 508, 50E, 519, 522, 524, 528, 52A, 52E, 533, 53F, 54A, 551, 55D, 562, 566, 56C, 573, 577, 57F, 582, 58G, 593, 599, 59B, 59F, 5A4, 5A6, 5AC, 5B5, 5C8, 5CE, 5D1, 5D3, 5EA, 5EE, 5EG, 5F9, 5G4, 5G6, 5GG, 607, 60D, 612, 618, 629, 62F, 632, 634, 63G, 649, 654, 65C, 66B, 678, 67E, 681, 683, 687, 689, 692, 69E, 6A3, 6A9, 6BA, 6BC, 6CB, 6CD, 6E1, 6E7, 6EF, 6F4, 6F8, 6FA, 6FE, 6G5, 6GB, 704, 706, 70G, 71D, 726, 72C, 737, 739, 73D, 73F, 748, 753, 755, 764, 766, 76C, 76G, 771, 77B, 782, 793, 7AA, 7AE, 7B3, 7BB, 7CA, 7CC, 7CG, 7D7, 7E6, 7E8, 7EC, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 814, 81A, 81C, 821, 825, 82B, 838, 83E, 841, 843, 849, 84D, 852, 85E, 863, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8B4, 8C5, 8CF, 8D6, 8DA, 8DG, 8E1, 8E7, 8FC, 8G7, 8G9, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 942, 948, 94E, 951, 953, 957, 95D, 964, 968, 96A, 96G, 979, 97B, 984, 98C, 98G, 99D, 9A6, 9B1, 9B3, 9B9, 9BD, 9BF, 9CE, 9DB, 9DF, 9E4, 9EC, 9F1, 9F5, 9G6, 9GE, A07, A0D, A12, A1A, A23, A2F, A3C, A3G, A45, A4B, A4D, A67, A69, A72, A7A, A7E, A8B, A8F, A96, AA1, AA7, AB2, AB6, ABC, ACF, AD8, ADA, AE9, AG1, AG5, AG7, B02, B08, B0C, B17, B1D, B24, B28, B2G, B44, B46, B4A, B4C, B57, B71, B73, B79, B7F, B84, B88, B8E, B8G, B9B, B9F, BAA, BAC, BB5, BB7, BC6, BC8, BD7, BDD, BEG, BFF, BG6, BGA, BGC, BGG, C01, C16, C1E, C29, C2F, C38, C3A, C3E, C43, C45, C4B, C54, C56, C61, C6B, C6D, C76, C83, C89, C8D, C92, ...|
|18|11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 10D, 115, 117, 11B, 11H, 127, 12D, 131, 135, 13B, 141, 145, 14D, 155, 157, 15H, 161, 167, 16B, 16H, 177, 17B, 17D, 17H, 18B, 191, 195, 19D, 19H, 1A5, 1AH, 1B1, 1C1, 1C7, 1CH, 1D5, 1DB, 1DD, 1E1, 1EB, 1EH, 1F5, 1F7, 1FD, 1G1, 1G5, 1G7, 1H1, 1HB, 1HD, 1HH, 205, 20B, 20D, 217, 21B, 21H, 227, 22H, 237, 23H, 247, 24D, 251, 255, 25D, 261, 265, 26D, 26H, 27D, 285, 28H, 291, 29B, 29D, 29H, 2A1, 2AB, 2B7, 2BB, 2BD, 2BH, 2CD, 2CH, 2D1, 2D5, 2E7, 2EB, 2F1, 2FB, 2G1, 2G5, 2GB, 2GH, 2HD, 2HH, 305, 30B, 311, 317, 321, 325, 32B, 32D, 335, 337, 33D, 345, 347, 34H, 351, 357, 367, 36B, 36D, 36H, 375, 37B, 381, 387, 38D, 39H, 3A1, 3AB, 3B1, 3BB, 3BH, 3C5, 3CD, 3D7, 3DB, 3DH, 3E5, 3E7, 3ED, 3F7, 3FH, 3GH, 3H1, 3H5, 3HB, 3HD, 401, 405, 407, 40B, 415, 417, 41D, 43B, 43H, 445, 44D, 45D, 465, 471, 475, 477, 47B, 47H, 487, 48B, 48D, 491, 49D, 4A5, 4A7, 4AB, 4AD, 4AH, 4B5, 4BH, 4CB, 4D1, 4DD, 4E1, 4E5, 4EB, 4F1, 4F5, 4FD, 4FH, 4GD, 4GH, 4H5, 4H7, 4HB, 4HH, 501, 507, 50H, 521, 527, 52B, 52D, 541, 545, 547, 54H, 55B, 55D, 565, 56D, 571, 577, 57D, 58D, 591, 595, 597, 5A1, 5AB, 5B5, 5BD, 5CB, 5D7, 5DD, 5DH, 5E1, 5E5, 5E7, 5EH, 5FB, 5FH, 5G5, 5H5, 5H7, 605, 607, 61B, 61H, 627, 62D, 62H, 631, 635, 63D, 641, 64B, 64D, 655, 661, 66B, 66H, 67B, 67D, 67H, 681, 68B, 695, 697, 6A5, 6A7, 6AD, 6AH, 6B1, 6BB, 6C1, 6D1, 6E7, 6EB, 6EH, 6F7, 6G5, 6G7, 6GB, 6H1, 6HH, 701, 705, 70D, 711, 717, 71B, 725, 727, 73B, 73H, 741, 747, 74B, 74H, 75D, 761, 765, 767, 76D, 76H, 775, 77H, 785, 78B, 797, 79B, 79H, 7AB, 7B1, 7B7, 7BB, 7D1, 7E1, 7EB, 7F1, 7F5, 7FB, 7FD, 7G1, 7H5, 7HH, 801, 80H, 817, 81B, 825, 831, 83B, 83D, 83H, 847, 84D, 851, 855, 857, 85B, 85H, 867, 86B, 86D, 871, 87B, 87D, 885, 88D, 88H, 89D, 8A5, 8AH, 8B1, 8B7, 8BB, 8BD, 8CB, 8D7, 8DB, 8DH, 8E7, 8ED, 8EH, 8FH, 8G7, 8GH, 8H5, 8HB, 901, 90B, 915, 921, 925, 92B, 92H, 931, 94B, 94D, 955, 95D, 95H, 96D, 96H, 977, 981, 987, 991, 995, 99B, 9AD, 9B5, 9B7, 9C5, 9DD, 9DH, 9E1, 9ED, 9F1, 9F5, 9FH, 9G5, 9GD, 9GH, 9H7, A0B, A0D, A0H, A11, A1D, A35, A37, A3D, A41, A47, A4B, A4H, A51, A5D, A5H, A6B, A6D, A75, A77, A85, A87, A95, A9B, AAD, ABB, AC1, AC5, AC7, ACB, ACD, ADH, AE7, AF1, AF7, AFH, AG1, AG5, AGB, AGD, AH1, AHB, AHD, B07, B0H, B11, B1B, B27, B2D, B2H, B35, ...|
|19|14, 1A, 1C, 1I, 23, 25, 29, 2F, 32, 34, 3A, 3E, 3G, 43, 47, 4D, 52, 56, 58, 5C, 5E, 5I, 6D, 6H, 74, 76, 7G, 7I, 85, 8B, 8F, 92, 98, 9A, A1, A3, A7, A9, B2, BE, BI, C1, C5, CB, CD, D4, DA, DG, E3, E5, EB, EF, EH, F8, G3, G7, G9, GD, H8, HE, I5, I7, IB, IH, 106, 10C, 10I, 113, 119, 11H, 122, 12A, 131, 133, 13D, 13F, 142, 146, 14C, 151, 155, 157, 15B, 164, 16C, 16G, 175, 179, 17F, 188, 18A, 199, 19F, 1A6, 1AC, 1AI, 1B1, 1B7, 1BH, 1C4, 1CA, 1CC, 1CI, 1D5, 1D9, 1DB, 1E4, 1EE, 1EG, 1F1, 1F7, 1FD, 1FF, 1G8, 1GC, 1GI, 1H7, 1HH, 1I6, 1IG, 205, 20B, 20H, 212, 21A, 21G, 221, 229, 22D, 238, 23I, 24B, 24D, 254, 256, 25A, 25C, 263, 26H, 272, 274, 278, 283, 287, 289, 28D, 29E, 29I, 2A7, 2AH, 2B6, 2BA, 2BG, 2C3, 2CH, 2D2, 2D8, 2DE, 2E3, 2E9, 2F2, 2F6, 2FC, 2FE, 2G5, 2G7, 2GD, 2H4, 2H6, 2HG, 2HI, 2I5, 304, 308, 30A, 30E, 311, 317, 31F, 322, 328, 33B, 33D, 344, 34C, 353, 359, 35F, 364, 36G, 371, 377, 37D, 37F, 382, 38E, 395, 3A4, 3A6, 3AA, 3AG, 3AI, 3B5, 3B9, 3BB, 3BF, 3C8, 3CA, 3CG, 3EC, 3EI, 3F5, 3FD, 3GC, 3H3, 3HH, 3I2, 3I4, 3I8, 3IE, 403, 407, 409, 40F, 418, 41I, 421, 425, 427, 42B, 42H, 43A, 443, 44B, 454, 45A, 45E, 461, 469, 46D, 472, 476, 481, 485, 48B, 48D, 48H, 494, 496, 49C, 4A3, 4B4, 4BA, 4BE, 4BG, 4D2, 4D6, 4D8, 4DI, 4EB, 4ED, 4F4, 4FC, 4FI, 4G5, 4GB, 4HA, 4HG, 4I1, 4I3, 4IF, 506, 50I, 517, 524, 52I, 535, 539, 53B, 53F, 53H, 548, 551, 557, 55D, 56C, 56E, 57B, 57D, 58G, 593, 59B, 59H, 5A2, 5A4, 5A8, 5AG, 5B3, 5BD, 5BF, 5C6, 5D1, 5DB, 5DH, 5EA, 5EC, 5EG, 5EI, 5F9, 5G2, 5G4, 5H1, 5H3, 5H9, 5HD, 5HF, 5I6, 5IE, 60D, 61I, 623, 629, 62H, 63E, 63G, 641, 649, 656, 658, 65C, 661, 667, 66D, 66H, 67A, 67C, 68F, 692, 694, 69A, 69E, 6A1, 6AF, 6B2, 6B6, 6B8, 6BE, 6BI, 6C5, 6CH, 6D4, 6DA, 6E5, 6E9, 6EF, 6F8, 6FG, 6G3, 6G7, 6HE, 6ID, 704, 70C, 70G, 713, 715, 71B, 72E, 737, 739, 746, 74E, 74I, 75B, 766, 76G, 76I, 773, 77B, 77H, 784, 788, 78A, 78E, 791, 799, 79D, 79F, 7A2, 7AC, 7AE, 7B5, 7BD, 7BH, 7CC, 7D3, 7DF, 7DH, 7E4, 7E8, 7EA, 7F7, 7G2, 7G6, 7GC, 7H1, 7H7, 7HB, 7IA, 7II, 809, 80F, 812, 81A, 821, 82D, 838, 83C, 83I, 845, 847, 85G, 85I, 869, 86H, 872, 87G, 881, 889, 892, 898, 8A1, 8A5, 8AB, 8BC, 8C3, 8C5, 8D2, 8E9, 8ED, 8EF, 8F8, 8FE, 8FI, 8GB, 8GH, 8H6, 8HA, 8HI, 902, 904, 908, 90A, 913, 92C, 92E, 931, 937, 93D, 93H, 944, 946, 94I, 953, 95F, 95H, 968, 96A, 977, 979, 986, 98C, 99D, 9AA, 9AI, 9B3, 9B5, 9B9, 9BB, 9CE, 9D3, 9DF, 9E2, 9EC, 9EE, 9EI, 9F5, 9F7, 9FD, 9G4, 9G6, 9GI, 9H9, 9HB, 9I2, 9IG, A03, A07, A0D, A12, ...|
|20|13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 109, 10J, 111, 11B, 11D, 11J, 123, 129, 12H, 131, 133, 137, 13J, 147, 14B, 14J, 153, 159, 161, 163, 171, 177, 17H, 183, 189, 18B, 18H, 197, 19D, 19J, 1A1, 1A7, 1AD, 1AH, 1AJ, 1BB, 1C1, 1C3, 1C7, 1CD, 1CJ, 1D1, 1DD, 1DH, 1E3, 1EB, 1F1, 1F9, 1FJ, 1G7, 1GD, 1GJ, 1H3, 1HB, 1HH, 1I1, 1I9, 1ID, 1J7, 1JH, 209, 20B, 211, 213, 217, 219, 21J, 22D, 22H, 22J, 233, 23H, 241, 243, 247, 257, 25B, 25J, 269, 26H, 271, 277, 27D, 287, 28B, 28H, 293, 29B, 29H, 2A9, 2AD, 2AJ, 2B1, 2BB, 2BD, 2BJ, 2C9, 2CB, 2D1, 2D3, 2D9, 2E7, 2EB, 2ED, 2EH, 2F3, 2F9, 2FH, 2G3, 2G9, 2HB, 2HD, 2I3, 2IB, 2J1, 2J7, 2JD, 301, 30D, 30H, 313, 319, 31B, 31H, 329, 32J, 33H, 33J, 343, 349, 34B, 34H, 351, 353, 357, 35J, 361, 367, 381, 387, 38D, 391, 39J, 3A9, 3B3, 3B7, 3B9, 3BD, 3BJ, 3C7, 3CB, 3CD, 3CJ, 3DB, 3E1, 3E3, 3E7, 3E9, 3ED, 3EJ, 3FB, 3G3, 3GB, 3H3, 3H9, 3HD, 3HJ, 3I7, 3IB, 3IJ, 3J3, 3JH, 401, 407, 409, 40D, 40J, 411, 417, 41H, 42H, 433, 437, 439, 44D, 44H, 44J, 459, 461, 463, 46D, 471, 477, 47D, 47J, 48H, 493, 497, 499, 4A1, 4AB, 4B3, 4BB, 4C7, 4D1, 4D7, 4DB, 4DD, 4DH, 4DJ, 4E9, 4F1, 4F7, 4FD, 4GB, 4GD, 4H9, 4HB, 4ID, 4IJ, 4J7, 4JD, 4JH, 4JJ, 503, 50B, 50H, 517, 519, 51J, 52D, 533, 539, 541, 543, 547, 549, 54J, 55B, 55D, 569, 56B, 56H, 571, 573, 57D, 581, 58J, 5A3, 5A7, 5AD, 5B1, 5BH, 5BJ, 5C3, 5CB, 5D7, 5D9, 5DD, 5E1, 5E7, 5ED, 5EH, 5F9, 5FB, 5GD, 5GJ, 5H1, 5H7, 5HB, 5HH, 5IB, 5IH, 5J1, 5J3, 5J9, 5JD, 5JJ, 60B, 60H, 613, 61H, 621, 627, 62J, 637, 63D, 63H, 653, 661, 66B, 66J, 673, 679, 67B, 67H, 68J, 69B, 69D, 6A9, 6AH, 6B1, 6BD, 6C7, 6CH, 6CJ, 6D3, 6DB, 6DH, 6E3, 6E7, 6E9, 6ED, 6EJ, 6F7, 6FB, 6FD, 6FJ, 6G9, 6GB, 6H1, 6H9, 6HD, 6I7, 6IH, 6J9, 6JB, 6JH, 701, 703, 70J, 71D, 71H, 723, 72B, 72H, 731, 73J, 747, 74H, 753, 759, 75H, 767, 76J, 77D, 77H, 783, 789, 78B, 79J, 7A1, 7AB, 7AJ, 7B3, 7BH, 7C1, 7C9, 7D1, 7D7, 7DJ, 7E3, 7E9, 7F9, 7FJ, 7G1, 7GH, 7I3, 7I7, 7I9, 7J1, 7J7, 7JB, 803, 809, 80H, 811, 819, 82B, 82D, 82H, 82J, 83B, 84J, 851, 857, 85D, 85J, 863, 869, 86B, 873, 877, 87J, 881, 88B, 88D, 899, 89B, 8A7, 8AD, 8BD, 8C9, 8CH, 8D1, 8D3, 8D7, 8D9, 8EB, 8EJ, 8FB, 8FH, 8G7, 8G9, 8GD, 8GJ, 8H1, 8H7, 8HH, 8HJ, 8IB, 8J1, 8J3, 8JD, 907, 90D, 90H, 913, 91B, ...|
|21|12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 102, 108, 10G, 10K, 111, 115, 11H, 124, 128, 12G, 12K, 135, 13H, 13J, 14G, 151, 15B, 15H, 162, 164, 16A, 16K, 175, 17B, 17D, 17J, 184, 188, 18A, 191, 19B, 19D, 19H, 1A2, 1A8, 1AA, 1B1, 1B5, 1BB, 1BJ, 1C8, 1CG, 1D5, 1DD, 1DJ, 1E4, 1E8, 1EG, 1F1, 1F5, 1FD, 1FH, 1GA, 1GK, 1HB, 1HD, 1I2, 1I4, 1I8, 1IA, 1IK, 1JD, 1JH, 1JJ, 1K2, 1KG, 1KK, 201, 205, 214, 218, 21G, 225, 22D, 22H, 232, 238, 241, 245, 24B, 24H, 254, 25A, 261, 265, 26B, 26D, 272, 274, 27A, 27K, 281, 28B, 28D, 28J, 29G, 29K, 2A1, 2A5, 2AB, 2AH, 2B4, 2BA, 2BG, 2CH, 2CJ, 2D8, 2DG, 2E5, 2EB, 2EH, 2F4, 2FG, 2FK, 2G5, 2GB, 2GD, 2GJ, 2HA, 2HK, 2IH, 2IJ, 2J2, 2J8, 2JA, 2JG, 2JK, 2K1, 2K5, 2KH, 2KJ, 304, 31H, 322, 328, 32G, 33D, 342, 34G, 34K, 351, 355, 35B, 35J, 362, 364, 36A, 371, 37B, 37D, 37H, 37J, 382, 388, 38K, 39B, 39J, 3AA, 3AG, 3AK, 3B5, 3BD, 3BH, 3C4, 3C8, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3E4, 3EA, 3EK, 3FJ, 3G4, 3G8, 3GA, 3HD, 3HH, 3HJ, 3I8, 3IK, 3J1, 3JB, 3JJ, 3K4, 3KA, 3KG, 40D, 40J, 412, 414, 41G, 425, 42H, 434, 43K, 44D, 44J, 452, 454, 458, 45A, 45K, 46B, 46H, 472, 47K, 481, 48H, 48J, 49K, 4A5, 4AD, 4AJ, 4B2, 4B4, 4B8, 4BG, 4C1, 4CB, 4CD, 4D2, 4DG, 4E5, 4EB, 4F2, 4F4, 4F8, 4FA, 4FK, 4GB, 4GD, 4H8, 4HA, 4HG, 4HK, 4I1, 4IB, 4IJ, 4JG, 4KJ, 502, 508, 50G, 51B, 51D, 51H, 524, 52K, 531, 535, 53D, 53J, 544, 548, 54K, 551, 562, 568, 56A, 56G, 56K, 575, 57J, 584, 588, 58A, 58G, 58K, 595, 59H, 5A2, 5A8, 5B1, 5B5, 5BB, 5C2, 5CA, 5CG, 5CK, 5E4, 5F1, 5FB, 5FJ, 5G2, 5G8, 5GA, 5GG, 5HH, 5I8, 5IA, 5J5, 5JD, 5JH, 5K8, 601, 60B, 60D, 60H, 614, 61A, 61G, 61K, 621, 625, 62B, 62J, 632, 634, 63A, 63K, 641, 64B, 64J, 652, 65G, 665, 66H, 66J, 674, 678, 67A, 685, 68J, 692, 698, 69G, 6A1, 6A5, 6B2, 6BA, 6BK, 6C5, 6CB, 6CJ, 6D8, 6DK, 6ED, 6EH, 6F2, 6F8, 6FA, 6GH, 6GJ, 6H8, 6HG, 6HK, 6ID, 6IH, 6J4, 6JG, 6K1, 6KD, 6KH, 702, 711, 71B, 71D, 728, 73D, 73H, 73J, 74A, 74G, 74K, 75B, 75H, 764, 768, 76G, 77H, 77J, 782, 784, 78G, 7A2, 7A4, 7AA, 7AG, 7B1, 7B5, 7BB, 7BD, 7C4, 7C8, 7CK, 7D1, 7DB, 7DD, 7E8, 7EA, 7F5, 7FB, 7GA, 7H5, 7HD, 7HH, 7HJ, 7I2, 7I4, 7J5, 7JD, 7K4, 7KA, 7KK, 801, 805, 80B, 80D, 80J, 818, 81A, 821, 82B, 82D, 832, 83G, 841, 845, 84B, 84J, ...|
|22|11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 107, 10F, 10J, 113, 11F, 11H, 12D, 12J, 137, 13D, 13J, 13L, 145, 14F, 14L, 155, 157, 15D, 15J, 161, 163, 16F, 173, 175, 179, 17F, 17L, 181, 18D, 18H, 191, 199, 19J, 1A5, 1AF, 1B1, 1B7, 1BD, 1BH, 1C3, 1C9, 1CD, 1CL, 1D3, 1DH, 1E5, 1EH, 1EJ, 1F7, 1F9, 1FD, 1FF, 1G3, 1GH, 1GL, 1H1, 1H5, 1HJ, 1I1, 1I3, 1I7, 1J5, 1J9, 1JH, 1K5, 1KD, 1KH, 1L1, 1L7, 1LL, 203, 209, 20F, 211, 217, 21J, 221, 227, 229, 22J, 22L, 235, 23F, 23H, 245, 247, 24D, 259, 25D, 25F, 25J, 263, 269, 26H, 271, 277, 287, 289, 28J, 295, 29F, 29L, 2A5, 2AD, 2B3, 2B7, 2BD, 2BJ, 2BL, 2C5, 2CH, 2D5, 2E1, 2E3, 2E7, 2ED, 2EF, 2EL, 2F3, 2F5, 2F9, 2FL, 2G1, 2G7, 2HJ, 2I3, 2I9, 2IH, 2JD, 2K1, 2KF, 2KJ, 2KL, 2L3, 2L9, 2LH, 2LL, 301, 307, 30J, 317, 319, 31D, 31F, 31J, 323, 32F, 335, 33D, 343, 349, 34D, 34J, 355, 359, 35H, 35L, 36D, 36H, 371, 373, 377, 37D, 37F, 37L, 389, 397, 39D, 39H, 39J, 3AL, 3B3, 3B5, 3BF, 3C5, 3C7, 3CH, 3D3, 3D9, 3DF, 3DL, 3EH, 3F1, 3F5, 3F7, 3FJ, 3G7, 3GJ, 3H5, 3HL, 3ID, 3IJ, 3J1, 3J3, 3J7, 3J9, 3JJ, 3K9, 3KF, 3KL, 3LH, 3LJ, 40D, 40F, 41F, 41L, 427, 42D, 42H, 42J, 431, 439, 43F, 443, 445, 44F, 457, 45H, 461, 46D, 46F, 46J, 46L, 479, 47L, 481, 48H, 48J, 493, 497, 499, 49J, 4A5, 4B1, 4C3, 4C7, 4CD, 4CL, 4DF, 4DH, 4DL, 4E7, 4F1, 4F3, 4F7, 4FF, 4FL, 4G5, 4G9, 4GL, 4H1, 4I1, 4I7, 4I9, 4IF, 4IJ, 4J3, 4JH, 4K1, 4K5, 4K7, 4KD, 4KH, 4L1, 4LD, 4LJ, 503, 50H, 50L, 515, 51H, 523, 529, 52D, 53H, 54D, 551, 559, 55D, 55J, 55L, 565, 575, 57H, 57J, 58D, 58L, 593, 59F, 5A7, 5AH, 5AJ, 5B1, 5B9, 5BF, 5BL, 5C3, 5C5, 5C9, 5CF, 5D1, 5D5, 5D7, 5DD, 5E1, 5E3, 5ED, 5EL, 5F3, 5FH, 5G5, 5GH, 5GJ, 5H3, 5H7, 5H9, 5I3, 5IH, 5IL, 5J5, 5JD, 5JJ, 5K1, 5KJ, 5L5, 5LF, 5LL, 605, 60D, 611, 61D, 625, 629, 62F, 62L, 631, 647, 649, 64J, 655, 659, 661, 665, 66D, 673, 679, 67L, 683, 689, 697, 69H, 69J, 6AD, 6BH, 6BL, 6C1, 6CD, 6CJ, 6D1, 6DD, 6DJ, 6E5, 6E9, 6EH, 6FH, 6FJ, 6G1, 6G3, 6GF, 6HL, 6I1, 6I7, 6ID, 6IJ, 6J1, 6J7, 6J9, 6JL, 6K3, 6KF, 6KH, 6L5, 6L7, 701, 703, 70J, 713, 721, 72H, 733, 737, 739, 73D, 73F, 74F, 751, 75D, 75J, 767, 769, 76D, 76J, 76L, 775, 77F, 77H, 787, 78H, 78J, 797, 79L, 7A5, 7A9, 7AF, 7B1, ...|
|23|16, 18, 1E, 1I, 1K, 21, 27, 2D, 2F, 2L, 32, 34, 3A, 3E, 3K, 45, 49, 4B, 4F, 4H, 4L, 5C, 5G, 5M, 61, 6B, 6D, 6J, 72, 76, 7C, 7I, 7K, 87, 89, 8D, 8F, 94, 9G, 9K, 9M, A3, A9, AB, AL, B4, BA, BG, BI, C1, C5, C7, CH, D8, DC, DE, DI, E9, EF, F2, F4, F8, FE, FM, G5, GB, GF, GL, H6, HA, HI, I5, I7, IH, IJ, J2, J6, JC, JK, K1, K3, K7, KJ, L4, L8, LG, LK, M3, MF, MH, 10C, 10I, 115, 11B, 11H, 11J, 122, 12C, 12I, 131, 133, 139, 13F, 13J, 13L, 14A, 14K, 14M, 153, 159, 15F, 15H, 166, 16A, 16G, 171, 17B, 17J, 186, 18E, 18K, 193, 197, 19F, 19L, 1A2, 1AA, 1AE, 1B5, 1BF, 1C4, 1C6, 1CG, 1CI, 1CM, 1D1, 1DB, 1E2, 1E6, 1E8, 1EC, 1F3, 1F7, 1F9, 1FD, 1GA, 1GE, 1GM, 1H9, 1HH, 1HL, 1I4, 1IA, 1J1, 1J5, 1JB, 1JH, 1K2, 1K8, 1KK, 1L1, 1L7, 1L9, 1LJ, 1LL, 1M4, 1ME, 1MG, 203, 205, 20B, 216, 21A, 21C, 21G, 21M, 225, 22D, 22J, 232, 241, 243, 24D, 24L, 258, 25E, 25K, 265, 26H, 26L, 274, 27A, 27C, 27I, 287, 28H, 29C, 29E, 29I, 2A1, 2A3, 2A9, 2AD, 2AF, 2AJ, 2B8, 2BA, 2BG, 2D4, 2DA, 2DG, 2E1, 2EJ, 2F6, 2FK, 2G1, 2G3, 2G7, 2GD, 2GL, 2H2, 2H4, 2HA, 2HM, 2I9, 2IB, 2IF, 2IH, 2IL, 2J4, 2JG, 2K5, 2KD, 2L2, 2L8, 2LC, 2LI, 2M3, 2M7, 2MF, 2MJ, 30A, 30E, 30K, 30M, 313, 319, 31B, 31H, 324, 331, 337, 33B, 33D, 34E, 34I, 34K, 357, 35J, 35L, 368, 36G, 36M, 375, 37B, 386, 38C, 38G, 38I, 397, 39H, 3A6, 3AE, 3B7, 3BL, 3C4, 3C8, 3CA, 3CE, 3CG, 3D3, 3DF, 3DL, 3E4, 3EM, 3F1, 3FH, 3FJ, 3GI, 3H1, 3H9, 3HF, 3HJ, 3HL, 3I2, 3IA, 3IG, 3J3, 3J5, 3JF, 3K6, 3KG, 3KM, 3LB, 3LD, 3LH, 3LJ, 3M6, 3MI, 3MK, 40D, 40F, 40L, 412, 414, 41E, 41M, 42H, 43I, 43M, 445, 44D, 456, 458, 45C, 45K, 46D, 46F, 46J, 474, 47A, 47G, 47K, 489, 48B, 49A, 49G, 49I, 4A1, 4A5, 4AB, 4B2, 4B8, 4BC, 4BE, 4BK, 4C1, 4C7, 4CJ, 4D2, 4D8, 4DM, 4E3, 4E9, 4EL, 4F6, 4FC, 4FG, 4GJ, 4HE, 4I1, 4I9, 4ID, 4IJ, 4IL, 4J4, 4K3, 4KF, 4KH, 4LA, 4LI, 4LM, 4MB, 502, 50C, 50E, 50I, 513, 519, 51F, 51J, 51L, 522, 528, 52G, 52K, 52M, 535, 53F, 53H, 544, 54C, 54G, 557, 55H, 566, 568, 56E, 56I, 56K, 57D, 584, 588, 58E, 58M, 595, 599, 5A4, 5AC, 5AM, 5B5, 5BB, 5BJ, 5C6, 5CI, 5D9, 5DD, 5DJ, 5E2, 5E4, 5F9, 5FB, 5FL, 5G6, 5GA, 5H1, 5H5, 5HD, 5I2, 5I8, 5IK, 5J1, 5J7, 5K4, 5KE, 5KG, 5L9, 5MC, 5MG, 5MI, 607, 60D, 60H, 616, 61C, 61K, 621, 629, 638, 63A, 63E, 63G, 645, 65A, 65C, 65I, 661, 667, 66B, 66H, 66J, 678, 67C, 681, 683, 68D, 68F, 698, 69A, 6A3, 6A9, 6B6, 6BM, 6C7, 6CB, 6CD, 6CH, 6CJ, 6DI, 6E3, 6EF, 6EL, 6F8, 6FA, 6FE, 6FK, 6FM, 6G5, 6GF, 6GH, 6H6, 6HG, 6HI, 6I5, 6IJ, 6J2, 6J6, 6JC, 6JK, 6K3, ...|
|24|15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 10H, 10N, 111, 117, 11D, 11H, 11J, 127, 12H, 12J, 12N, 135, 13B, 13D, 141, 145, 14B, 14J, 155, 15D, 15N, 167, 16D, 16J, 16N, 177, 17D, 17H, 181, 185, 18J, 195, 19H, 19J, 1A5, 1A7, 1AB, 1AD, 1AN, 1BD, 1BH, 1BJ, 1BN, 1CD, 1CH, 1CJ, 1CN, 1DJ, 1DN, 1E7, 1EH, 1F1, 1F5, 1FB, 1FH, 1G7, 1GB, 1GH, 1GN, 1H7, 1HD, 1I1, 1I5, 1IB, 1ID, 1IN, 1J1, 1J7, 1JH, 1JJ, 1K5, 1K7, 1KD, 1L7, 1LB, 1LD, 1LH, 1LN, 1M5, 1MD, 1MJ, 1N1, 1NN, 201, 20B, 20J, 215, 21B, 21H, 221, 22D, 22H, 22N, 235, 237, 23D, 241, 24B, 255, 257, 25B, 25H, 25J, 261, 265, 267, 26B, 26N, 271, 277, 28H, 28N, 295, 29D, 2A7, 2AH, 2B7, 2BB, 2BD, 2BH, 2BN, 2C7, 2CB, 2CD, 2CJ, 2D7, 2DH, 2DJ, 2DN, 2E1, 2E5, 2EB, 2EN, 2FB, 2FJ, 2G7, 2GD, 2GH, 2GN, 2H7, 2HB, 2HJ, 2HN, 2ID, 2IH, 2IN, 2J1, 2J5, 2JB, 2JD, 2JJ, 2K5, 2L1, 2L7, 2LB, 2LD, 2MD, 2MH, 2MJ, 2N5, 2NH, 2NJ, 305, 30D, 30J, 311, 317, 321, 327, 32B, 32D, 331, 33B, 33N, 347, 34N, 35D, 35J, 35N, 361, 365, 367, 36H, 375, 37B, 37H, 38B, 38D, 395, 397, 3A5, 3AB, 3AJ, 3B1, 3B5, 3B7, 3BB, 3BJ, 3C1, 3CB, 3CD, 3CN, 3DD, 3DN, 3E5, 3EH, 3EJ, 3EN, 3F1, 3FB, 3FN, 3G1, 3GH, 3GJ, 3H1, 3H5, 3H7, 3HH, 3I1, 3IJ, 3JJ, 3JN, 3K5, 3KD, 3L5, 3L7, 3LB, 3LJ, 3MB, 3MD, 3MH, 3N1, 3N7, 3ND, 3NH, 405, 407, 415, 41B, 41D, 41J, 41N, 425, 42J, 431, 435, 437, 43D, 43H, 43N, 44B, 44H, 44N, 45D, 45H, 45N, 46B, 46J, 471, 475, 487, 491, 49B, 49J, 49N, 4A5, 4A7, 4AD, 4BB, 4BN, 4C1, 4CH, 4D1, 4D5, 4DH, 4E7, 4EH, 4EJ, 4EN, 4F7, 4FD, 4FJ, 4FN, 4G1, 4G5, 4GB, 4GJ, 4GN, 4H1, 4H7, 4HH, 4HJ, 4I5, 4ID, 4IH, 4J7, 4JH, 4K5, 4K7, 4KD, 4KH, 4KJ, 4LB, 4M1, 4M5, 4MB, 4MJ, 4N1, 4N5, 4NN, 507, 50H, 50N, 515, 51D, 51N, 52B, 531, 535, 53B, 53H, 53J, 54N, 551, 55B, 55J, 55N, 56D, 56H, 571, 57D, 57J, 587, 58B, 58H, 59D, 59N, 5A1, 5AH, 5BJ, 5BN, 5C1, 5CD, 5CJ, 5CN, 5DB, 5DH, 5E1, 5E5, 5ED, 5FB, 5FD, 5FH, 5FJ, 5G7, 5HB, 5HD, 5HJ, 5I1, 5I7, 5IB, 5IH, 5IJ, 5J7, 5JB, 5JN, 5K1, 5KB, 5KD, 5L5, 5L7, 5LN, 5M5, 5N1, 5NH, 601, 605, 607, 60B, 60D, 61B, 61J, 627, 62D, 62N, 631, 635, 63B, 63D, 63J, 645, 647, 64J, 655, 657, 65H, 667, 66D, 66H, 66N, 677, 67D, ...|
|25|14, 16, 1C, 1G, 1I, 1M, 23, 29, 2B, 2H, 2L, 2N, 34, 38, 3E, 3M, 41, 43, 47, 49, 4D, 52, 56, 5C, 5E, 5O, 61, 67, 6D, 6H, 6N, 74, 76, 7G, 7I, 7M, 7O, 8B, 8N, 92, 94, 98, 9E, 9G, A1, A7, AD, AJ, AL, B2, B6, B8, BI, C7, CB, CD, CH, D6, DC, DM, DO, E3, E9, EH, EN, F4, F8, FE, FM, G1, G9, GJ, GL, H6, H8, HE, HI, HO, I7, IB, ID, IH, J4, JC, JG, JO, K3, K9, KL, KN, LG, LM, M7, MD, MJ, ML, N2, NC, NI, NO, O1, O7, OD, OH, OJ, 106, 10G, 10I, 10M, 113, 119, 11B, 11N, 122, 128, 12G, 131, 139, 13J, 142, 148, 14E, 14I, 151, 157, 15B, 15J, 15N, 16C, 16M, 179, 17B, 17L, 17N, 182, 184, 18E, 193, 197, 199, 19D, 1A2, 1A6, 1A8, 1AC, 1B7, 1BB, 1BJ, 1C4, 1CC, 1CG, 1CM, 1D3, 1DH, 1DL, 1E2, 1E8, 1EG, 1EM, 1F9, 1FD, 1FJ, 1FL, 1G6, 1G8, 1GE, 1GO, 1H1, 1HB, 1HD, 1HJ, 1IC, 1IG, 1II, 1IM, 1J3, 1J9, 1JH, 1JN, 1K4, 1L1, 1L3, 1LD, 1LL, 1M6, 1MC, 1MI, 1N1, 1ND, 1NH, 1NN, 1O4, 1O6, 1OC, 1OO, 209, 212, 214, 218, 21E, 21G, 21M, 221, 223, 227, 22J, 22L, 232, 24B, 24H, 24N, 256, 25O, 269, 26N, 272, 274, 278, 27E, 27M, 281, 283, 289, 28L, 296, 298, 29C, 29E, 29I, 29O, 2AB, 2AN, 2B6, 2BI, 2BO, 2C3, 2C9, 2CH, 2CL, 2D4, 2D8, 2DM, 2E1, 2E7, 2E9, 2ED, 2EJ, 2EL, 2F2, 2FC, 2G7, 2GD, 2GH, 2GJ, 2HI, 2HM, 2HO, 2I9, 2IL, 2IN, 2J8, 2JG, 2JM, 2K3, 2K9, 2L2, 2L8, 2LC, 2LE, 2M1, 2MB, 2MN, 2N6, 2NM, 2OB, 2OH, 2OL, 2ON, 302, 304, 30E, 311, 317, 31D, 326, 328, 32O, 331, 33N, 344, 34C, 34I, 34M, 34O, 353, 35B, 35H, 362, 364, 36E, 373, 37D, 37J, 386, 388, 38C, 38E, 38O, 39B, 39D, 3A4, 3A6, 3AC, 3AG, 3AI, 3B3, 3BB, 3C4, 3D3, 3D7, 3DD, 3DL, 3EC, 3EE, 3EI, 3F1, 3FH, 3FJ, 3FN, 3G6, 3GC, 3GI, 3GM, 3H9, 3HB, 3I8, 3IE, 3IG, 3IM, 3J1, 3J7, 3JL, 3K2, 3K6, 3K8, 3KE, 3KI, 3KO, 3LB, 3LH, 3LN, 3MC, 3MG, 3MM, 3N9, 3NH, 3NN, 3O2, 403, 40L, 416, 41E, 41I, 41O, 421, 427, 434, 43G, 43I, 449, 44H, 44L, 458, 45M, 467, 469, 46D, 46L, 472, 478, 47C, 47E, 47I, 47O, 487, 48B, 48D, 48J, 494, 496, 49G, 49O, 4A3, 4AH, 4B2, 4BE, 4BG, 4BM, 4C1, 4C3, 4CJ, 4D8, 4DC, 4DI, 4E1, 4E7, 4EB, 4F4, 4FC, 4FM, 4G3, 4G9, 4GH, 4H2, 4HE, 4I3, 4I7, 4ID, 4IJ, 4IL, 4JO, 4K1, 4KB, 4KJ, 4KN, 4LC, 4LG, 4LO, 4MB, 4MH, 4N4, 4N8, 4NE, 4O9, 4OJ, 4OL, 50C, 51D, 51H, 51J, 526, 52C, 52G, 533, 539, 53H, 53L, 544, 551, 553, 557, 559, 55L, 56O, 571, 577, 57D, 57J, 57N, 584, 586, 58I, 58M, 599, 59B, 59L, 59N, 5AE, 5AG, 5B7, 5BD, 5C8, 5CO, 5D7, 5DB, 5DD, 5DH, 5DJ, 5EG, 5EO, 5FB, 5FH, 5G2, 5G4, 5G8, 5GE, 5GG, 5GM, 5H7, 5H9, 5HL, 5I6, 5I8, 5II, 5J7, 5JD, 5JH, 5JN, 5K6, 5KC, ...|
|26|13, 15, 1B, 1F, 1H, 1L, 21, 27, 29, 2F, 2J, 2L, 31, 35, 3B, 3J, 3N, 3P, 43, 45, 49, 4N, 51, 57, 59, 5J, 5L, 61, 67, 6B, 6H, 6N, 6P, 79, 7B, 7F, 7H, 83, 8F, 8J, 8L, 8P, 95, 97, 9H, 9N, A3, A9, AB, AH, AL, AN, B7, BL, BP, C1, C5, CJ, CP, D9, DB, DF, DL, E3, E9, EF, EJ, EP, F7, FB, FJ, G3, G5, GF, GH, GN, H1, H7, HF, HJ, HL, HP, IB, IJ, IN, J5, J9, JF, K1, K3, KL, L1, LB, LH, LN, LP, M5, MF, ML, N1, N3, N9, NF, NJ, NL, O7, OH, OJ, ON, P3, P9, PB, PN, 101, 107, 10F, 10P, 117, 11H, 11P, 125, 12B, 12F, 12N, 133, 137, 13F, 13J, 147, 14H, 153, 155, 15F, 15H, 15L, 15N, 167, 16L, 16P, 171, 175, 17J, 17N, 17P, 183, 18N, 191, 199, 19J, 1A1, 1A5, 1AB, 1AH, 1B5, 1B9, 1BF, 1BL, 1C3, 1C9, 1CL, 1CP, 1D5, 1D7, 1DH, 1DJ, 1DP, 1E9, 1EB, 1EL, 1EN, 1F3, 1FL, 1FP, 1G1, 1G5, 1GB, 1GH, 1GP, 1H5, 1HB, 1I7, 1I9, 1IJ, 1J1, 1JB, 1JH, 1JN, 1K5, 1KH, 1KL, 1L1, 1L7, 1L9, 1LF, 1M1, 1MB, 1N3, 1N5, 1N9, 1NF, 1NH, 1NN, 1O1, 1O3, 1O7, 1OJ, 1OL, 1P1, 209, 20F, 20L, 213, 21L, 225, 22J, 22N, 22P, 233, 239, 23H, 23L, 23N, 243, 24F, 24P, 251, 255, 257, 25B, 25H, 263, 26F, 26N, 279, 27F, 27J, 27P, 287, 28B, 28J, 28N, 29B, 29F, 29L, 29N, 2A1, 2A7, 2A9, 2AF, 2AP, 2BJ, 2BP, 2C3, 2C5, 2D3, 2D7, 2D9, 2DJ, 2E5, 2E7, 2EH, 2EP, 2F5, 2FB, 2FH, 2G9, 2GF, 2GJ, 2GL, 2H7, 2HH, 2I3, 2IB, 2J1, 2JF, 2JL, 2JP, 2K1, 2K5, 2K7, 2KH, 2L3, 2L9, 2LF, 2M7, 2M9, 2MP, 2N1, 2NN, 2O3, 2OB, 2OH, 2OL, 2ON, 2P1, 2P9, 2PF, 2PP, 301, 30B, 30P, 319, 31F, 321, 323, 327, 329, 32J, 335, 337, 33N, 33P, 345, 349, 34B, 34L, 353, 35L, 36J, 36N, 373, 37B, 381, 383, 387, 38F, 395, 397, 39B, 39J, 39P, 3A5, 3A9, 3AL, 3AN, 3BJ, 3BP, 3C1, 3C7, 3CB, 3CH, 3D5, 3DB, 3DF, 3DH, 3DN, 3E1, 3E7, 3EJ, 3EP, 3F5, 3FJ, 3FN, 3G3, 3GF, 3GN, 3H3, 3H7, 3I7, 3IP, 3J9, 3JH, 3JL, 3K1, 3K3, 3K9, 3L5, 3LH, 3LJ, 3M9, 3MH, 3ML, 3N7, 3NL, 3O5, 3O7, 3OB, 3OJ, 3OP, 3P5, 3P9, 3PB, 3PF, 3PL, 403, 407, 409, 40F, 40P, 411, 41B, 41J, 41N, 42B, 42L, 437, 439, 43F, 43J, 43L, 44B, 44P, 453, 459, 45H, 45N, 461, 46J, 471, 47B, 47H, 47N, 485, 48F, 491, 49F, 49J, 49P, 4A5, 4A7, 4B9, 4BB, 4BL, 4C3, 4C7, 4CL, 4CP, 4D7, 4DJ, 4DP, 4EB, 4EF, 4EL, 4FF, 4FP, 4G1, 4GH, 4HH, 4HL, 4HN, 4I9, 4IF, 4IJ, 4J5, 4JB, 4JJ, 4JN, 4K5, 4L1, 4L3, 4L7, 4L9, 4LL, 4MN, 4MP, 4N5, 4NB, 4NH, 4NL, 4O1, 4O3, 4OF, 4OJ, 4P5, 4P7, 4PH, 4PJ, 509, 50B, 511, 517, 521, 52H, 52P, 533, 535, 539, 53B, 547, 54F, 551, 557, 55H, 55J, 55N, 563, 565, 56B, 56L, 56N, 579, 57J, 57L, 585, 58J, 58P, 593, 599, 59H, 59N, ...|
|27|12, 14, 1A, 1E, 1G, 1K, 1Q, 25, 27, 2D, 2H, 2J, 2P, 32, 38, 3G, 3K, 3M, 3Q, 41, 45, 4J, 4N, 52, 54, 5E, 5G, 5M, 61, 65, 6B, 6H, 6J, 72, 74, 78, 7A, 7M, 87, 8B, 8D, 8H, 8N, 8P, 98, 9E, 9K, 9Q, A1, A7, AB, AD, AN, BA, BE, BG, BK, C7, CD, CN, CP, D2, D8, DG, DM, E1, E5, EB, EJ, EN, F4, FE, FG, FQ, G1, G7, GB, GH, GP, H2, H4, H8, HK, I1, I5, ID, IH, IN, J8, JA, K1, K7, KH, KN, L2, L4, LA, LK, LQ, M5, M7, MD, MJ, MN, MP, NA, NK, NM, NQ, O5, OB, OD, OP, P2, P8, PG, PQ, Q7, QH, QP, 104, 10A, 10E, 10M, 111, 115, 11D, 11H, 124, 12E, 12Q, 131, 13B, 13D, 13H, 13J, 142, 14G, 14K, 14M, 14Q, 15D, 15H, 15J, 15N, 16G, 16K, 171, 17B, 17J, 17N, 182, 188, 18M, 18Q, 195, 19B, 19J, 19P, 1AA, 1AE, 1AK, 1AM, 1B5, 1B7, 1BD, 1BN, 1BP, 1C8, 1CA, 1CG, 1D7, 1DB, 1DD, 1DH, 1DN, 1E2, 1EA, 1EG, 1EM, 1FH, 1FJ, 1G2, 1GA, 1GK, 1GQ, 1H5, 1HD, 1HP, 1I2, 1I8, 1IE, 1IG, 1IM, 1J7, 1JH, 1K8, 1KA, 1KE, 1KK, 1KM, 1L1, 1L5, 1L7, 1LB, 1LN, 1LP, 1M4, 1NB, 1NH, 1NN, 1O4, 1OM, 1P5, 1PJ, 1PN, 1PP, 1Q2, 1Q8, 1QG, 1QK, 1QM, 201, 20D, 20N, 20P, 212, 214, 218, 21E, 21Q, 22B, 22J, 234, 23A, 23E, 23K, 241, 245, 24D, 24H, 254, 258, 25E, 25G, 25K, 25Q, 261, 267, 26H, 27A, 27G, 27K, 27M, 28J, 28N, 28P, 298, 29K, 29M, 2A5, 2AD, 2AJ, 2AP, 2B4, 2BM, 2C1, 2C5, 2C7, 2CJ, 2D2, 2DE, 2DM, 2EB, 2EP, 2F4, 2F8, 2FA, 2FE, 2FG, 2FQ, 2GB, 2GH, 2GN, 2HE, 2HG, 2I5, 2I7, 2J2, 2J8, 2JG, 2JM, 2JQ, 2K1, 2K5, 2KD, 2KJ, 2L2, 2L4, 2LE, 2M1, 2MB, 2MH, 2N2, 2N4, 2N8, 2NA, 2NK, 2O5, 2O7, 2ON, 2OP, 2P4, 2P8, 2PA, 2PK, 2Q1, 2QJ, 30G, 30K, 30Q, 317, 31N, 31P, 322, 32A, 32Q, 331, 335, 33D, 33J, 33P, 342, 34E, 34G, 35B, 35H, 35J, 35P, 362, 368, 36M, 371, 375, 377, 37D, 37H, 37N, 388, 38E, 38K, 397, 39B, 39H, 3A2, 3AA, 3AG, 3AK, 3BJ, 3CA, 3CK, 3D1, 3D5, 3DB, 3DD, 3DJ, 3EE, 3EQ, 3F1, 3FH, 3FP, 3G2, 3GE, 3H1, 3HB, 3HD, 3HH, 3HP, 3I4, 3IA, 3IE, 3IG, 3IK, 3IQ, 3J7, 3JB, 3JD, 3JJ, 3K2, 3K4, 3KE, 3KM, 3KQ, 3LD, 3LN, 3M8, 3MA, 3MG, 3MK, 3MM, 3NB, 3NP, 3O2, 3O8, 3OG, 3OM, 3OQ, 3PH, 3PP, 3Q8, 3QE, 3QK, 401, 40B, 40N, 41A, 41E, 41K, 41Q, 421, 432, 434, 43E, 43M, 43Q, 44D, 44H, 44P, 45A, 45G, 461, 465, 46B, 474, 47E, 47G, 485, 494, 498, 49A, 49M, 4A1, 4A5, 4AH, 4AN, 4B4, 4B8, 4BG, 4CB, 4CD, 4CH, 4CJ, 4D4, 4E5, 4E7, 4ED, 4EJ, 4EP, 4F2, 4F8, 4FA, 4FM, 4FQ, 4GB, 4GD, 4GN, 4GP, 4HE, 4HG, 4I5, 4IB, 4J4, 4JK, 4K1, 4K5, 4K7, 4KB, 4KD, 4L8, 4LG, 4M1, 4M7, 4MH, 4MJ, 4MN, 4N2, 4N4, 4NA, 4NK, 4NM, 4O7, 4OH, 4OJ, 4P2, 4PG, 4PM, 4PQ, 4Q5, 4QD, 4QJ, ...|
|28|11, 13, 19, 1D, 1F, 1J, 1P, 23, 25, 2B, 2F, 2H, 2N, 2R, 35, 3D, 3H, 3J, 3N, 3P, 41, 4F, 4J, 4P, 4R, 59, 5B, 5H, 5N, 5R, 65, 6B, 6D, 6N, 6P, 71, 73, 7F, 7R, 83, 85, 89, 8F, 8H, 8R, 95, 9B, 9H, 9J, 9P, A1, A3, AD, AR, B3, B5, B9, BN, C1, CB, CD, CH, CN, D3, D9, DF, DJ, DP, E5, E9, EH, ER, F1, FB, FD, FJ, FN, G1, G9, GD, GF, GJ, H3, HB, HF, HN, HR, I5, IH, IJ, J9, JF, JP, K3, K9, KB, KH, KR, L5, LB, LD, LJ, LP, M1, M3, MF, MP, MR, N3, N9, NF, NH, O1, O5, OB, OJ, P1, P9, PJ, PR, Q5, QB, QF, QN, R1, R5, RD, RH, 103, 10D, 10P, 10R, 119, 11B, 11F, 11H, 11R, 12D, 12H, 12J, 12N, 139, 13D, 13F, 13J, 14B, 14F, 14N, 155, 15D, 15H, 15N, 161, 16F, 16J, 16P, 173, 17B, 17H, 181, 185, 18B, 18D, 18N, 18P, 193, 19D, 19F, 19P, 19R, 1A5, 1AN, 1AR, 1B1, 1B5, 1BB, 1BH, 1BP, 1C3, 1C9, 1D3, 1D5, 1DF, 1DN, 1E5, 1EB, 1EH, 1EP, 1F9, 1FD, 1FJ, 1FP, 1FR, 1G5, 1GH, 1GR, 1HH, 1HJ, 1HN, 1I1, 1I3, 1I9, 1ID, 1IF, 1IJ, 1J3, 1J5, 1JB, 1KH, 1KN, 1L1, 1L9, 1LR, 1M9, 1MN, 1MR, 1N1, 1N5, 1NB, 1NJ, 1NN, 1NP, 1O3, 1OF, 1OP, 1OR, 1P3, 1P5, 1P9, 1PF, 1PR, 1QB, 1QJ, 1R3, 1R9, 1RD, 1RJ, 1RR, 203, 20B, 20F, 211, 215, 21B, 21D, 21H, 21N, 21P, 223, 22D, 235, 23B, 23F, 23H, 24D, 24H, 24J, 251, 25D, 25F, 25P, 265, 26B, 26H, 26N, 27D, 27J, 27N, 27P, 289, 28J, 293, 29B, 29R, 2AD, 2AJ, 2AN, 2AP, 2B1, 2B3, 2BD, 2BP, 2C3, 2C9, 2CR, 2D1, 2DH, 2DJ, 2ED, 2EJ, 2ER, 2F5, 2F9, 2FB, 2FF, 2FN, 2G1, 2GB, 2GD, 2GN, 2H9, 2HJ, 2HP, 2I9, 2IB, 2IF, 2IH, 2IR, 2JB, 2JD, 2K1, 2K3, 2K9, 2KD, 2KF, 2KP, 2L5, 2LN, 2MJ, 2MN, 2N1, 2N9, 2NP, 2NR, 2O3, 2OB, 2OR, 2P1, 2P5, 2PD, 2PJ, 2PP, 2Q1, 2QD, 2QF, 2R9, 2RF, 2RH, 2RN, 2RR, 305, 30J, 30P, 311, 313, 319, 31D, 31J, 323, 329, 32F, 331, 335, 33B, 33N, 343, 349, 34D, 35B, 361, 36B, 36J, 36N, 371, 373, 379, 383, 38F, 38H, 395, 39D, 39H, 3A1, 3AF, 3AP, 3AR, 3B3, 3BB, 3BH, 3BN, 3BR, 3C1, 3C5, 3CB, 3CJ, 3CN, 3CP, 3D3, 3DD, 3DF, 3DP, 3E5, 3E9, 3EN, 3F5, 3FH, 3FJ, 3FP, 3G1, 3G3, 3GJ, 3H5, 3H9, 3HF, 3HN, 3I1, 3I5, 3IN, 3J3, 3JD, 3JJ, 3JP, 3K5, 3KF, 3KR, 3LD, 3LH, 3LN, 3M1, 3M3, 3N3, 3N5, 3NF, 3NN, 3NR, 3OD, 3OH, 3OP, 3P9, 3PF, 3PR, 3Q3, 3Q9, 3R1, 3RB, 3RD, 401, 40R, 413, 415, 41H, 41N, 41R, 42B, 42H, 42P, 431, 439, 443, 445, 449, 44B, 44N, 45N, 45P, 463, 469, 46F, 46J, 46P, 46R, 47B, 47F, 47R, 481, 48B, 48D, 491, 493, 49J, 49P, 4AH, 4B5, 4BD, 4BH, 4BJ, 4BN, 4BP, 4CJ, 4CR, 4DB, 4DH, 4DR, 4E1, 4E5, 4EB, 4ED, 4EJ, 4F1, 4F3, 4FF, 4FP, 4FR, 4G9, 4GN, 4H1, 4H5, 4HB, 4HJ, 4HP, ...|
|29|12, 18, 1C, 1E, 1I, 1O, 21, 23, 29, 2D, 2F, 2L, 2P, 32, 3A, 3E, 3G, 3K, 3M, 3Q, 4B, 4F, 4L, 4N, 54, 56, 5C, 5I, 5M, 5S, 65, 67, 6H, 6J, 6N, 6P, 78, 7K, 7O, 7Q, 81, 87, 89, 8J, 8P, 92, 98, 9A, 9G, 9K, 9M, A3, AH, AL, AN, AR, BC, BI, BS, C1, C5, CB, CJ, CP, D2, D6, DC, DK, DO, E3, ED, EF, EP, ER, F4, F8, FE, FM, FQ, FS, G3, GF, GN, GR, H6, HA, HG, HS, I1, IJ, IP, J6, JC, JI, JK, JQ, K7, KD, KJ, KL, KR, L4, L8, LA, LM, M3, M5, M9, MF, ML, MN, N6, NA, NG, NO, O5, OD, ON, P2, P8, PE, PI, PQ, Q3, Q7, QF, QJ, R4, RE, RQ, RS, S9, SB, SF, SH, SR, 10C, 10G, 10I, 10M, 117, 11B, 11D, 11H, 128, 12C, 12K, 131, 139, 13D, 13J, 13P, 14A, 14E, 14K, 14Q, 155, 15B, 15N, 15R, 164, 166, 16G, 16I, 16O, 175, 177, 17H, 17J, 17P, 18E, 18I, 18K, 18O, 191, 197, 19F, 19L, 19R, 1AK, 1AM, 1B3, 1BB, 1BL, 1BR, 1C4, 1CC, 1CO, 1CS, 1D5, 1DB, 1DD, 1DJ, 1E2, 1EC, 1F1, 1F3, 1F7, 1FD, 1FF, 1FL, 1FP, 1FR, 1G2, 1GE, 1GG, 1GM, 1HR, 1I4, 1IA, 1II, 1J7, 1JH, 1K2, 1K6, 1K8, 1KC, 1KI, 1KQ, 1L1, 1L3, 1L9, 1LL, 1M2, 1M4, 1M8, 1MA, 1ME, 1MK, 1N3, 1NF, 1NN, 1O6, 1OC, 1OG, 1OM, 1P1, 1P5, 1PD, 1PH, 1Q2, 1Q6, 1QC, 1QE, 1QI, 1QO, 1QQ, 1R3, 1RD, 1S4, 1SA, 1SE, 1SG, 20B, 20F, 20H, 20R, 21A, 21C, 21M, 221, 227, 22D, 22J, 238, 23E, 23I, 23K, 243, 24D, 24P, 254, 25K, 265, 26B, 26F, 26H, 26L, 26N, 274, 27G, 27M, 27S, 28H, 28J, 296, 298, 2A1, 2A7, 2AF, 2AL, 2AP, 2AR, 2B2, 2BA, 2BG, 2BQ, 2BS, 2C9, 2CN, 2D4, 2DA, 2DM, 2DO, 2DS, 2E1, 2EB, 2EN, 2EP, 2FC, 2FE, 2FK, 2FO, 2FQ, 2G7, 2GF, 2H4, 2HS, 2I3, 2I9, 2IH, 2J4, 2J6, 2JA, 2JI, 2K5, 2K7, 2KB, 2KJ, 2KP, 2L2, 2L6, 2LI, 2LK, 2MD, 2MJ, 2ML, 2MR, 2N2, 2N8, 2NM, 2NS, 2O3, 2O5, 2OB, 2OF, 2OL, 2P4, 2PA, 2PG, 2Q1, 2Q5, 2QB, 2QN, 2R2, 2R8, 2RC, 2S9, 2SR, 308, 30G, 30K, 30Q, 30S, 315, 31R, 32A, 32C, 32S, 337, 33B, 33N, 348, 34I, 34K, 34O, 353, 359, 35F, 35J, 35L, 35P, 362, 36A, 36E, 36G, 36M, 373, 375, 37F, 37N, 37R, 38C, 38M, 395, 397, 39D, 39H, 39J, 3A6, 3AK, 3AO, 3B1, 3B9, 3BF, 3BJ, 3C8, 3CG, 3CQ, 3D3, 3D9, 3DH, 3DR, 3EA, 3EO, 3ES, 3F5, 3FB, 3FD, 3GC, 3GE, 3GO, 3H3, 3H7, 3HL, 3HP, 3I4, 3IG, 3IM, 3J5, 3J9, 3JF, 3K6, 3KG, 3KI, 3L5, 3M2, 3M6, 3M8, 3MK, 3MQ, 3N1, 3ND, 3NJ, 3NR, 3O2, 3OA, 3P3, 3P5, 3P9, 3PB, 3PN, 3QM, 3QO, 3R1, 3R7, 3RD, 3RH, 3RN, 3RP, 3S8, 3SC, 3SO, 3SQ, 407, 409, 40P, 40R, 41E, 41K, 42B, 42R, 436, 43A, 43C, 43G, 43I, 44B, 44J, 452, 458, 45I, 45K, 45O, 461, 463, 469, 46J, 46L, 474, 47E, 47G, 47Q, 48B, 48H, 48L, 48R, 496, 49C, 49I, ...|
|30|11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 107, 10B, 10J, 10T, 117, 11B, 11H, 11N, 127, 12B, 12H, 12N, 131, 137, 13J, 13N, 13T, 141, 14B, 14D, 14J, 14T, 151, 15B, 15D, 15J, 167, 16B, 16D, 16H, 16N, 16T, 177, 17D, 17J, 18B, 18D, 18N, 191, 19B, 19H, 19N, 1A1, 1AD, 1AH, 1AN, 1AT, 1B1, 1B7, 1BJ, 1BT, 1CH, 1CJ, 1CN, 1CT, 1D1, 1D7, 1DB, 1DD, 1DH, 1DT, 1E1, 1E7, 1FB, 1FH, 1FN, 1G1, 1GJ, 1GT, 1HD, 1HH, 1HJ, 1HN, 1HT, 1I7, 1IB, 1ID, 1IJ, 1J1, 1JB, 1JD, 1JH, 1JJ, 1JN, 1JT, 1KB, 1KN, 1L1, 1LD, 1LJ, 1LN, 1LT, 1M7, 1MB, 1MJ, 1MN, 1N7, 1NB, 1NH, 1NJ, 1NN, 1NT, 1O1, 1O7, 1OH, 1P7, 1PD, 1PH, 1PJ, 1QD, 1QH, 1QJ, 1QT, 1RB, 1RD, 1RN, 1S1, 1S7, 1SD, 1SJ, 1T7, 1TD, 1TH, 1TJ, 201, 20B, 20N, 211, 21H, 221, 227, 22B, 22D, 22H, 22J, 22T, 23B, 23H, 23N, 24B, 24D, 24T, 251, 25N, 25T, 267, 26D, 26H, 26J, 26N, 271, 277, 27H, 27J, 27T, 28D, 28N, 28T, 29B, 29D, 29H, 29J, 29T, 2AB, 2AD, 2AT, 2B1, 2B7, 2BB, 2BD, 2BN, 2C1, 2CJ, 2DD, 2DH, 2DN, 2E1, 2EH, 2EJ, 2EN, 2F1, 2FH, 2FJ, 2FN, 2G1, 2G7, 2GD, 2GH, 2GT, 2H1, 2HN, 2HT, 2I1, 2I7, 2IB, 2IH, 2J1, 2J7, 2JB, 2JD, 2JJ, 2JN, 2JT, 2KB, 2KH, 2KN, 2L7, 2LB, 2LH, 2LT, 2M7, 2MD, 2MH, 2ND, 2O1, 2OB, 2OJ, 2ON, 2OT, 2P1, 2P7, 2PT, 2QB, 2QD, 2QT, 2R7, 2RB, 2RN, 2S7, 2SH, 2SJ, 2SN, 2T1, 2T7, 2TD, 2TH, 2TJ, 2TN, 2TT, 307, 30B, 30D, 30J, 30T, 311, 31B, 31J, 31N, 327, 32H, 32T, 331, 337, 33B, 33D, 33T, 34D, 34H, 34N, 351, 357, 35B, 35T, 367, 36H, 36N, 36T, 377, 37H, 37T, 38D, 38H, 38N, 38T, 391, 39T, 3A1, 3AB, 3AJ, 3AN, 3B7, 3BB, 3BJ, 3C1, 3C7, 3CJ, 3CN, 3CT, 3DJ, 3DT, 3E1, 3EH, 3FD, 3FH, 3FJ, 3G1, 3G7, 3GB, 3GN, 3GT, 3H7, 3HB, 3HJ, 3IB, 3ID, 3IH, 3IJ, 3J1, 3JT, 3K1, 3K7, 3KD, 3KJ, 3KN, 3KT, 3L1, 3LD, 3LH, 3LT, 3M1, 3MB, 3MD, 3MT, 3N1, 3NH, 3NN, 3OD, 3OT, 3P7, 3PB, 3PD, 3PH, 3PJ, 3QB, 3QJ, 3R1, 3R7, 3RH, 3RJ, 3RN, 3RT, 3S1, 3S7, 3SH, 3SJ, 3T1, 3TB, 3TD, 3TN, 407, 40D, 40H, 40N, 411, 417, 41D, ...|
|31|16, 1A, 1C, 1G, 1M, 1S, 1U, 25, 29, 2B, 2H, 2L, 2R, 34, 38, 3A, 3E, 3G, 3K, 43, 47, 4D, 4F, 4P, 4R, 52, 58, 5C, 5I, 5O, 5Q, 65, 67, 6B, 6D, 6P, 76, 7A, 7C, 7G, 7M, 7O, 83, 89, 8F, 8L, 8N, 8T, 92, 94, 9E, 9S, A1, A3, A7, AL, AR, B6, B8, BC, BI, BQ, C1, C7, CB, CH, CP, CT, D6, DG, DI, DS, DU, E5, E9, EF, EN, ER, ET, F2, FE, FM, FQ, G3, G7, GD, GP, GR, HE, HK, HU, I5, IB, ID, IJ, IT, J4, JA, JC, JI, JO, JS, JU, KB, KL, KN, KR, L2, L8, LA, LM, LQ, M1, M9, MJ, MR, N6, NE, NK, NQ, NU, O7, OD, OH, OP, OT, PC, PM, Q3, Q5, QF, QH, QL, QN, R2, RG, RK, RM, RQ, S9, SD, SF, SJ, T8, TC, TK, TU, U7, UB, UH, UN, 106, 10A, 10G, 10M, 10U, 115, 11H, 11L, 11R, 11T, 128, 12A, 12G, 12Q, 12S, 137, 139, 13F, 142, 146, 148, 14C, 14I, 14O, 151, 157, 15D, 164, 166, 16G, 16O, 173, 179, 17F, 17N, 184, 188, 18E, 18K, 18M, 18S, 199, 19J, 1A6, 1A8, 1AC, 1AI, 1AK, 1AQ, 1AU, 1B1, 1B5, 1BH, 1BJ, 1BP, 1CS, 1D3, 1D9, 1DH, 1E4, 1EE, 1ES, 1F1, 1F3, 1F7, 1FD, 1FL, 1FP, 1FR, 1G2, 1GE, 1GO, 1GQ, 1GU, 1H1, 1H5, 1HB, 1HN, 1I4, 1IC, 1IO, 1IU, 1J3, 1J9, 1JH, 1JL, 1JT, 1K2, 1KG, 1KK, 1KQ, 1KS, 1L1, 1L7, 1L9, 1LF, 1LP, 1ME, 1MK, 1MO, 1MQ, 1NJ, 1NN, 1NP, 1O4, 1OG, 1OI, 1OS, 1P5, 1PB, 1PH, 1PN, 1QA, 1QG, 1QK, 1QM, 1R3, 1RD, 1RP, 1S2, 1SI, 1T1, 1T7, 1TB, 1TD, 1TH, 1TJ, 1TT, 1UA, 1UG, 1UM, 209, 20B, 20R, 20T, 21K, 21Q, 223, 229, 22D, 22F, 22J, 22R, 232, 23C, 23E, 23O, 247, 24H, 24N, 254, 256, 25A, 25C, 25M, 263, 265, 26L, 26N, 26T, 272, 274, 27E, 27M, 289, 292, 296, 29C, 29K, 2A5, 2A7, 2AB, 2AJ, 2B4, 2B6, 2BA, 2BI, 2BO, 2BU, 2C3, 2CF, 2CH, 2D8, 2DE, 2DG, 2DM, 2DQ, 2E1, 2EF, 2EL, 2EP, 2ER, 2F2, 2F6, 2FC, 2FO, 2FU, 2G5, 2GJ, 2GN, 2GT, 2HA, 2HI, 2HO, 2HS, 2IN, 2JA, 2JK, 2JS, 2K1, 2K7, 2K9, 2KF, 2L6, 2LI, 2LK, 2M5, 2MD, 2MH, 2MT, 2NC, 2NM, 2NO, 2NS, 2O5, 2OB, 2OH, 2OL, 2ON, 2OR, 2P2, 2PA, 2PE, 2PG, 2PM, 2Q1, 2Q3, 2QD, 2QL, 2QP, 2R8, 2RI, 2RU, 2S1, 2S7, 2SB, 2SD, 2ST, 2TC, 2TG, 2TM, 2TU, 2U5, 2U9, 2UR, 304, 30E, 30K, 30Q, 313, 31D, 31P, 328, 32C, 32I, 32O, 32Q, 33N, 33P, 344, 34C, 34G, 34U, 353, 35B, 35N, 35T, 36A, 36E, 36K, 379, 37J, 37L, 386, 391, 395, 397, 39J, 39P, 39T, 3AA, 3AG, 3AO, 3AS, 3B5, 3BR, 3BT, 3C2, 3C4, 3CG, 3DD, 3DF, 3DL, 3DR, 3E2, 3E6, 3EC, 3EE, 3EQ, 3EU, 3FB, 3FD, 3FN, 3FP, 3GA, 3GC, 3GS, 3H3, 3HN, 3I8, 3IG, 3IK, 3IM, 3IQ, 3IS, 3JJ, 3JR, 3K8, 3KE, 3KO, 3KQ, 3KU, 3L5, 3L7, 3LD, 3LN, 3LP, 3M6, 3MG, 3MI, 3MS, 3NB, 3NH, 3NL, 3NR, 3O4, 3OA, 3OG, 3P1, ...|
|32|15, 19, 1B, 1F, 1L, 1R, 1T, 23, 27, 29, 2F, 2J, 2P, 31, 35, 37, 3B, 3D, 3H, 3V, 43, 49, 4B, 4L, 4N, 4T, 53, 57, 5D, 5J, 5L, 5V, 61, 65, 67, 6J, 6V, 73, 75, 79, 7F, 7H, 7R, 81, 87, 8D, 8F, 8L, 8P, 8R, 95, 9J, 9N, 9P, 9T, AB, AH, AR, AT, B1, B7, BF, BL, BR, BV, C5, CD, CH, CP, D3, D5, DF, DH, DN, DR, E1, E9, ED, EF, EJ, EV, F7, FB, FJ, FN, FT, G9, GB, GT, H3, HD, HJ, HP, HR, I1, IB, IH, IN, IP, IV, J5, J9, JB, JN, K1, K3, K7, KD, KJ, KL, L1, L5, LB, LJ, LT, M5, MF, MN, MT, N3, N7, NF, NL, NP, O1, O5, OJ, OT, P9, PB, PL, PN, PR, PT, Q7, QL, QP, QR, QV, RD, RH, RJ, RN, SB, SF, SN, T1, T9, TD, TJ, TP, U7, UB, UH, UN, UV, V5, VH, VL, VR, VT, 107, 109, 10F, 10P, 10R, 115, 117, 11D, 11V, 123, 125, 129, 12F, 12L, 12T, 133, 139, 13V, 141, 14B, 14J, 14T, 153, 159, 15H, 15T, 161, 167, 16D, 16F, 16L, 171, 17B, 17T, 17V, 183, 189, 18B, 18H, 18L, 18N, 18R, 197, 199, 19F, 1AH, 1AN, 1AT, 1B5, 1BN, 1C1, 1CF, 1CJ, 1CL, 1CP, 1CV, 1D7, 1DB, 1DD, 1DJ, 1DV, 1E9, 1EB, 1EF, 1EH, 1EL, 1ER, 1F7, 1FJ, 1FR, 1G7, 1GD, 1GH, 1GN, 1GV, 1H3, 1HB, 1HF, 1HT, 1I1, 1I7, 1I9, 1ID, 1IJ, 1IL, 1IR, 1J5, 1JP, 1JV, 1K3, 1K5, 1KT, 1L1, 1L3, 1LD, 1LP, 1LR, 1M5, 1MD, 1MJ, 1MP, 1MV, 1NH, 1NN, 1NR, 1NT, 1O9, 1OJ, 1OV, 1P7, 1PN, 1Q5, 1QB, 1QF, 1QH, 1QL, 1QN, 1R1, 1RD, 1RJ, 1RP, 1SB, 1SD, 1ST, 1SV, 1TL, 1TR, 1U3, 1U9, 1UD, 1UF, 1UJ, 1UR, 1V1, 1VB, 1VD, 1VN, 205, 20F, 20L, 211, 213, 217, 219, 21J, 21V, 221, 22H, 22J, 22P, 22T, 22V, 239, 23H, 243, 24R, 24V, 255, 25D, 25T, 25V, 263, 26B, 26R, 26T, 271, 279, 27F, 27L, 27P, 285, 287, 28T, 293, 295, 29B, 29F, 29L, 2A3, 2A9, 2AD, 2AF, 2AL, 2AP, 2AV, 2BB, 2BH, 2BN, 2C5, 2C9, 2CF, 2CR, 2D3, 2D9, 2DD, 2E7, 2EP, 2F3, 2FB, 2FF, 2FL, 2FN, 2FT, 2GJ, 2GV, 2H1, 2HH, 2HP, 2HT, 2I9, 2IN, 2J1, 2J3, 2J7, 2JF, 2JL, 2JR, 2JV, 2K1, 2K5, 2KB, 2KJ, 2KN, 2KP, 2KV, 2L9, 2LB, 2LL, 2LT, 2M1, 2MF, 2MP, 2N5, 2N7, 2ND, 2NH, 2NJ, 2O3, 2OH, 2OL, 2OR, 2P3, 2P9, 2PD, 2PV, 2Q7, 2QH, 2QN, 2QT, 2R5, 2RF, 2RR, 2S9, 2SD, 2SJ, 2SP, 2SR, 2TN, 2TP, 2U3, 2UB, 2UF, 2UT, 2V1, 2V9, 2VL, 2VR, 307, 30B, 30H, 315, 31F, 31H, 321, 32R, 32V, 331, 33D, 33J, 33N, 343, 349, 34H, 34L, 34T, 35J, 35L, 35P, 35R, 367, 373, 375, 37B, 37H, 37N, 37R, 381, 383, 38F, 38J, 38V, 391, 39B, 39D, 39T, 39V, 3AF, 3AL, 3B9, 3BP, 3C1, 3C5, 3C7, 3CB, 3CD, 3D3, 3DB, 3DN, 3DT, 3E7, 3E9, 3ED, 3EJ, 3EL, 3ER, 3F5, 3F7, 3FJ, 3FT, 3FV, 3G9, 3GN, 3GT, 3H1, 3H7, 3HF, 3HL, 3HR, 3IB, ...|
|33|14, 18, 1A, 1E, 1K, 1Q, 1S, 21, 25, 27, 2D, 2H, 2N, 2V, 32, 34, 38, 3A, 3E, 3S, 3W, 45, 47, 4H, 4J, 4P, 4V, 52, 58, 5E, 5G, 5Q, 5S, 5W, 61, 6D, 6P, 6T, 6V, 72, 78, 7A, 7K, 7Q, 7W, 85, 87, 8D, 8H, 8J, 8T, 9A, 9E, 9G, 9K, A1, A7, AH, AJ, AN, AT, B4, BA, BG, BK, BQ, C1, C5, CD, CN, CP, D2, D4, DA, DE, DK, DS, DW, E1, E5, EH, EP, ET, F4, F8, FE, FQ, FS, GD, GJ, GT, H2, H8, HA, HG, HQ, HW, I5, I7, ID, IJ, IN, IP, J4, JE, JG, JK, JQ, JW, K1, KD, KH, KN, KV, L8, LG, LQ, M1, M7, MD, MH, MP, MV, N2, NA, NE, NS, O5, OH, OJ, OT, OV, P2, P4, PE, PS, PW, Q1, Q5, QJ, QN, QP, QT, RG, RK, RS, S5, SD, SH, SN, ST, TA, TE, TK, TQ, U1, U7, UJ, UN, UT, UV, V8, VA, VG, VQ, VS, W5, W7, WD, WV, 102, 104, 108, 10E, 10K, 10S, 111, 117, 11T, 11V, 128, 12G, 12Q, 12W, 135, 13D, 13P, 13T, 142, 148, 14A, 14G, 14S, 155, 15N, 15P, 15T, 162, 164, 16A, 16E, 16G, 16K, 16W, 171, 177, 188, 18E, 18K, 18S, 19D, 19N, 1A4, 1A8, 1AA, 1AE, 1AK, 1AS, 1AW, 1B1, 1B7, 1BJ, 1BT, 1BV, 1C2, 1C4, 1C8, 1CE, 1CQ, 1D5, 1DD, 1DP, 1DV, 1E2, 1E8, 1EG, 1EK, 1ES, 1EW, 1FD, 1FH, 1FN, 1FP, 1FT, 1G2, 1G4, 1GA, 1GK, 1H7, 1HD, 1HH, 1HJ, 1IA, 1IE, 1IG, 1IQ, 1J5, 1J7, 1JH, 1JP, 1JV, 1K4, 1KA, 1KS, 1L1, 1L5, 1L7, 1LJ, 1LT, 1M8, 1MG, 1MW, 1ND, 1NJ, 1NN, 1NP, 1NT, 1NV, 1O8, 1OK, 1OQ, 1OW, 1PH, 1PJ, 1Q2, 1Q4, 1QQ, 1QW, 1R7, 1RD, 1RH, 1RJ, 1RN, 1RV, 1S4, 1SE, 1SG, 1SQ, 1T7, 1TH, 1TN, 1U2, 1U4, 1U8, 1UA, 1UK, 1UW, 1V1, 1VH, 1VJ, 1VP, 1VT, 1VV, 1W8, 1WG, 201, 20P, 20T, 212, 21A, 21Q, 21S, 21W, 227, 22N, 22P, 22T, 234, 23A, 23G, 23K, 23W, 241, 24N, 24T, 24V, 254, 258, 25E, 25S, 261, 265, 267, 26D, 26H, 26N, 272, 278, 27E, 27S, 27W, 285, 28H, 28P, 28V, 292, 29S, 2AD, 2AN, 2AV, 2B2, 2B8, 2BA, 2BG, 2C5, 2CH, 2CJ, 2D2, 2DA, 2DE, 2DQ, 2E7, 2EH, 2EJ, 2EN, 2EV, 2F4, 2FA, 2FE, 2FG, 2FK, 2FQ, 2G1, 2G5, 2G7, 2GD, 2GN, 2GP, 2H2, 2HA, 2HE, 2HS, 2I5, 2IH, 2IJ, 2IP, 2IT, 2IV, 2JE, 2JS, 2JW, 2K5, 2KD, 2KJ, 2KN, 2L8, 2LG, 2LQ, 2LW, 2M5, 2MD, 2MN, 2N2, 2NG, 2NK, 2NQ, 2NW, 2O1, 2OT, 2OV, 2P8, 2PG, 2PK, 2Q1, 2Q5, 2QD, 2QP, 2QV, 2RA, 2RE, 2RK, 2S7, 2SH, 2SJ, 2T2, 2TS, 2TW, 2U1, 2UD, 2UJ, 2UN, 2V2, 2V8, 2VG, 2VK, 2VS, 2WH, 2WJ, 2WN, 2WP, 304, 30W, 311, 317, 31D, 31J, 31N, 31T, 31V, 32A, 32E, 32Q, 32S, 335, 337, 33N, 33P, 348, 34E, 351, 35H, 35P, 35T, 35V, 362, 364, 36Q, 371, 37D, 37J, 37T, 37V, 382, 388, 38A, 38G, 38Q, 38S, 397, 39H, 39J, 39T, 3AA, 3AG, 3AK, 3AQ, 3B1, 3B7, 3BD, 3BT, ...|
|34|13, 17, 19, 1D, 1J, 1P, 1R, 1X, 23, 25, 2B, 2F, 2L, 2T, 2X, 31, 35, 37, 3B, 3P, 3T, 41, 43, 4D, 4F, 4L, 4R, 4V, 53, 59, 5B, 5L, 5N, 5R, 5T, 67, 6J, 6N, 6P, 6T, 71, 73, 7D, 7J, 7P, 7V, 7X, 85, 89, 8B, 8L, 91, 95, 97, 9B, 9P, 9V, A7, A9, AD, AJ, AR, AX, B5, B9, BF, BN, BR, C1, CB, CD, CN, CP, CV, D1, D7, DF, DJ, DL, DP, E3, EB, EF, EN, ER, EX, FB, FD, FV, G3, GD, GJ, GP, GR, GX, H9, HF, HL, HN, HT, I1, I5, I7, IJ, IT, IV, J1, J7, JD, JF, JR, JV, K3, KB, KL, KT, L5, LD, LJ, LP, LT, M3, M9, MD, ML, MP, N5, NF, NR, NT, O5, O7, OB, OD, ON, P3, P7, P9, PD, PR, PV, PX, Q3, QN, QR, R1, RB, RJ, RN, RT, S1, SF, SJ, SP, SV, T5, TB, TN, TR, TX, U1, UB, UD, UJ, UT, UV, V7, V9, VF, VX, W3, W5, W9, WF, WL, WT, X1, X7, XT, XV, 107, 10F, 10P, 10V, 113, 11B, 11N, 11R, 11X, 125, 127, 12D, 12P, 131, 13J, 13L, 13P, 13V, 13X, 145, 149, 14B, 14F, 14R, 14T, 151, 161, 167, 16D, 16L, 175, 17F, 17T, 17X, 181, 185, 18B, 18J, 18N, 18P, 18V, 199, 19J, 19L, 19P, 19R, 19V, 1A3, 1AF, 1AR, 1B1, 1BD, 1BJ, 1BN, 1BT, 1C3, 1C7, 1CF, 1CJ, 1CX, 1D3, 1D9, 1DB, 1DF, 1DL, 1DN, 1DT, 1E5, 1EP, 1EV, 1F1, 1F3, 1FR, 1FV, 1FX, 1G9, 1GL, 1GN, 1GX, 1H7, 1HD, 1HJ, 1HP, 1I9, 1IF, 1IJ, 1IL, 1IX, 1J9, 1JL, 1JT, 1KB, 1KP, 1KV, 1L1, 1L3, 1L7, 1L9, 1LJ, 1LV, 1M3, 1M9, 1MR, 1MT, 1NB, 1ND, 1O1, 1O7, 1OF, 1OL, 1OP, 1OR, 1OV, 1P5, 1PB, 1PL, 1PN, 1PX, 1QD, 1QN, 1QT, 1R7, 1R9, 1RD, 1RF, 1RP, 1S3, 1S5, 1SL, 1SN, 1ST, 1SX, 1T1, 1TB, 1TJ, 1U3, 1UR, 1UV, 1V3, 1VB, 1VR, 1VT, 1VX, 1W7, 1WN, 1WP, 1WT, 1X3, 1X9, 1XF, 1XJ, 1XV, 1XX, 20L, 20R, 20T, 211, 215, 21B, 21P, 21V, 221, 223, 229, 22D, 22J, 22V, 233, 239, 23N, 23R, 23X, 24B, 24J, 24P, 24T, 25L, 265, 26F, 26N, 26R, 26X, 271, 277, 27T, 287, 289, 28P, 28X, 293, 29F, 29T, 2A5, 2A7, 2AB, 2AJ, 2AP, 2AV, 2B1, 2B3, 2B7, 2BD, 2BL, 2BP, 2BR, 2BX, 2C9, 2CB, 2CL, 2CT, 2CX, 2DD, 2DN, 2E1, 2E3, 2E9, 2ED, 2EF, 2EV, 2FB, 2FF, 2FL, 2FT, 2G1, 2G5, 2GN, 2GV, 2H7, 2HD, 2HJ, 2HR, 2I3, 2IF, 2IT, 2IX, 2J5, 2JB, 2JD, 2K7, 2K9, 2KJ, 2KR, 2KV, 2LB, 2LF, 2LN, 2M1, 2M7, 2MJ, 2MN, 2MT, 2NF, 2NP, 2NR, 2O9, 2P1, 2P5, 2P7, 2PJ, 2PP, 2PT, 2Q7, 2QD, 2QL, 2QP, 2QX, 2RL, 2RN, 2RR, 2RT, 2S7, 2T1, 2T3, 2T9, 2TF, 2TL, 2TP, 2TV, 2TX, 2UB, 2UF, 2UR, 2UT, 2V5, 2V7, 2VN, 2VP, 2W7, 2WD, 2WX, 2XF, 2XN, 2XR, 2XT, 2XX, 301, 30N, 30V, 319, 31F, 31P, 31R, 31V, 323, 325, 32B, 32L, 32N, 331, 33B, 33D, 33N, 343, 349, 34D, 34J, 34R, 34X, 355, 35L, ...|
|35|12, 16, 18, 1C, 1I, 1O, 1Q, 1W, 21, 23, 29, 2D, 2J, 2R, 2V, 2X, 32, 34, 38, 3M, 3Q, 3W, 3Y, 49, 4B, 4H, 4N, 4R, 4X, 54, 56, 5G, 5I, 5M, 5O, 61, 6D, 6H, 6J, 6N, 6T, 6V, 76, 7C, 7I, 7O, 7Q, 7W, 81, 83, 8D, 8R, 8V, 8X, 92, 9G, 9M, 9W, 9Y, A3, A9, AH, AN, AT, AX, B4, BC, BG, BO, BY, C1, CB, CD, CJ, CN, CT, D2, D6, D8, DC, DO, DW, E1, E9, ED, EJ, EV, EX, FG, FM, FW, G3, G9, GB, GH, GR, GX, H4, H6, HC, HI, HM, HO, I1, IB, ID, IH, IN, IT, IV, J8, JC, JI, JQ, K1, K9, KJ, KR, KX, L4, L8, LG, LM, LQ, LY, M3, MH, MR, N4, N6, NG, NI, NM, NO, NY, OD, OH, OJ, ON, P2, P6, P8, PC, PW, Q1, Q9, QJ, QR, QV, R2, R8, RM, RQ, RW, S3, SB, SH, ST, SX, T4, T6, TG, TI, TO, TY, U1, UB, UD, UJ, V2, V6, V8, VC, VI, VO, VW, W3, W9, WV, WX, X8, XG, XQ, XW, Y3, YB, YN, YR, YX, 104, 106, 10C, 10O, 10Y, 11H, 11J, 11N, 11T, 11V, 122, 126, 128, 12C, 12O, 12Q, 12W, 13V, 142, 148, 14G, 14Y, 159, 15N, 15R, 15T, 15X, 164, 16C, 16G, 16I, 16O, 171, 17B, 17D, 17H, 17J, 17N, 17T, 186, 18I, 18Q, 193, 199, 19D, 19J, 19R, 19V, 1A4, 1A8, 1AM, 1AQ, 1AW, 1AY, 1B3, 1B9, 1BB, 1BH, 1BR, 1CC, 1CI, 1CM, 1CO, 1DD, 1DH, 1DJ, 1DT, 1E6, 1E8, 1EI, 1EQ, 1EW, 1F3, 1F9, 1FR, 1FX, 1G2, 1G4, 1GG, 1GQ, 1H3, 1HB, 1HR, 1I6, 1IC, 1IG, 1II, 1IM, 1IO, 1IY, 1JB, 1JH, 1JN, 1K6, 1K8, 1KO, 1KQ, 1LD, 1LJ, 1LR, 1LX, 1M2, 1M4, 1M8, 1MG, 1MM, 1MW, 1MY, 1N9, 1NN, 1NX, 1O4, 1OG, 1OI, 1OM, 1OO, 1OY, 1PB, 1PD, 1PT, 1PV, 1Q2, 1Q6, 1Q8, 1QI, 1QQ, 1R9, 1RX, 1S2, 1S8, 1SG, 1SW, 1SY, 1T3, 1TB, 1TR, 1TT, 1TX, 1U6, 1UC, 1UI, 1UM, 1UY, 1V1, 1VN, 1VT, 1VV, 1W2, 1W6, 1WC, 1WQ, 1WW, 1X1, 1X3, 1X9, 1XD, 1XJ, 1XV, 1Y2, 1Y8, 1YM, 1YQ, 1YW, 209, 20H, 20N, 20R, 21I, 221, 22B, 22J, 22N, 22T, 22V, 232, 23O, 241, 243, 24J, 24R, 24V, 258, 25M, 25W, 25Y, 263, 26B, 26H, 26N, 26R, 26T, 26X, 274, 27C, 27G, 27I, 27O, 27Y, 281, 28B, 28J, 28N, 292, 29C, 29O, 29Q, 29W, 2A1, 2A3, 2AJ, 2AX, 2B2, 2B8, 2BG, 2BM, 2BQ, 2C9, 2CH, 2CR, 2CX, 2D4, 2DC, 2DM, 2DY, 2ED, 2EH, 2EN, 2ET, 2EV, 2FO, 2FQ, 2G1, 2G9, 2GD, 2GR, 2GV, 2H4, 2HG, 2HM, 2HY, 2I3, 2I9, 2IT, 2J4, 2J6, 2JM, 2KD, 2KH, 2KJ, 2KV, 2L2, 2L6, 2LI, 2LO, 2LW, 2M1, 2M9, 2MV, 2MX, 2N2, 2N4, 2NG, 2O9, 2OB, 2OH, 2ON, 2OT, 2OX, 2P4, 2P6, 2PI, 2PM, 2PY, 2Q1, 2QB, 2QD, 2QT, 2QV, 2RC, 2RI, 2S3, 2SJ, 2SR, 2SV, 2SX, 2T2, 2T4, 2TQ, 2TY, 2UB, 2UH, 2UR, 2UT, 2UX, 2V4, 2V6, 2VC, 2VM, 2VO, 2W1, 2WB, 2WD, 2WN, 2X2, 2X8, 2XC, 2XI, 2XQ, 2XW, 2Y3, 2YJ, ...|
|36|11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 101, 105, 107, 10B, 10N, 10P, 10V, 11T, 11Z, 125, 12D, 12V, 135, 13J, 13N, 13P, 13T, 13Z, 147, 14B, 14D, 14J, 14V, 155, 157, 15B, 15D, 15H, 15N, 15Z, 16B, 16J, 16V, 171, 175, 17B, 17J, 17N, 17V, 17Z, 18D, 18H, 18N, 18P, 18T, 18Z, 191, 197, 19H, 1A1, 1A7, 1AB, 1AD, 1B1, 1B5, 1B7, 1BH, 1BT, 1BV, 1C5, 1CD, 1CJ, 1CP, 1CV, 1DD, 1DJ, 1DN, 1DP, 1E1, 1EB, 1EN, 1EV, 1FB, 1FP, 1FV, 1FZ, 1G1, 1G5, 1G7, 1GH, 1GT, 1GZ, 1H5, 1HN, 1HP, 1I5, 1I7, 1IT, 1IZ, 1J7, 1JD, 1JH, 1JJ, 1JN, 1JV, 1K1, 1KB, 1KD, 1KN, 1L1, 1LB, 1LH, 1LT, 1LV, 1LZ, 1M1, 1MB, 1MN, 1MP, 1N5, 1N7, 1ND, 1NH, 1NJ, 1NT, 1O1, 1OJ, 1P7, 1PB, 1PH, 1PP, 1Q5, 1Q7, 1QB, 1QJ, 1QZ, 1R1, 1R5, 1RD, 1RJ, 1RP, 1RT, 1S5, 1S7, 1ST, 1SZ, 1T1, 1T7, 1TB, 1TH, 1TV, 1U1, 1U5, 1U7, 1UD, 1UH, 1UN, 1UZ, 1V5, 1VB, 1VP, 1VT, 1VZ, 1WB, 1WJ, 1WP, 1WT, 1XJ, 1Y1, 1YB, 1YJ, 1YN, 1YT, 1YV, 1Z1, 1ZN, 1ZZ, 201, 20H, 20P, 20T, 215, 21J, 21T, 21V, 21Z, 227, 22D, 22J, 22N, 22P, 22T, 22Z, 237, 23B, 23D, 23J, 23T, 23V, 245, 24D, 24H, 24V, 255, 25H, 25J, 25P, 25T, 25V, 26B, 26P, 26T, 26Z, 277, 27D, 27H, 27Z, 287, 28H, 28N, 28T, 291, 29B, 29N, 2A1, 2A5, 2AB, 2AH, 2AJ, 2BB, 2BD, 2BN, 2BV, 2BZ, 2CD, 2CH, 2CP, 2D1, 2D7, 2DJ, 2DN, 2DT, 2ED, 2EN, 2EP, 2F5, 2FV, 2FZ, 2G1, 2GD, 2GJ, 2GN, 2GZ, 2H5, 2HD, 2HH, 2HP, 2IB, 2ID, 2IH, 2IJ, 2IV, 2JN, 2JP, 2JV, 2K1, 2K7, 2KB, 2KH, 2KJ, 2KV, 2KZ, 2LB, 2LD, 2LN, 2LP, 2M5, 2M7, 2MN, 2MT, 2ND, 2NT, 2O1, 2O5, 2O7, 2OB, 2OD, 2OZ, 2P7, 2PJ, 2PP, 2PZ, 2Q1, 2Q5, 2QB, 2QD, 2QJ, 2QT, 2QV, 2R7, 2RH, 2RJ, 2RT, 2S7, 2SD, 2SH, 2SN, 2SV, 2T1, 2T7, 2TN, ...|

Addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) are the basic operations of arithmetic (https://en.wikipedia.org/wiki/Arithmetic, https://www.rieselprime.de/ziki/Arithmetic, https://mathworld.wolfram.com/Arithmetic.html) (which is also the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html)). In the addition operation, the identity element (https://en.wikipedia.org/wiki/Identity_element, https://mathworld.wolfram.com/IdentityElement.html) is 0, and all natural numbers > 0 can be written as the sum of many 1’s, and the number 1 cannot be broken up; in the multiplication operation, the identity element is 1, and all natural numbers > 1 can be written as the product of many prime numbers, and the prime numbers cannot be broken up. Also, prime numbers are central in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html) because of the fundamental theorem of arithmetic (https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic, https://t5k.org/glossary/xpage/FundamentalTheorem.html, https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html, http://www.numericana.com/answer/primes.htm#fta): every natural number greater than 1 is either a prime itself or can be factorized (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) as a product of primes that is unique up to (https://en.wikipedia.org/wiki/Up_to) their order. Also, primes are the natural numbers *n* > 1 such that if *n* divides (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) *x*×*y* (*x* and *y* are natural numbers), then *n* divides either *x* or *y* (or both). Also, prime numbers are the numbers *n* such that the ring (https://en.wikipedia.org/wiki/Ring_(mathematics), https://mathworld.wolfram.com/Ring.html) of integers modulo *n* (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html) (i.e. the ring *Z*<sub>*n*</sub>) is a field (https://en.wikipedia.org/wiki/Field_(mathematics), https://mathworld.wolfram.com/Field.html) (also is an integral domain (https://en.wikipedia.org/wiki/Integral_domain, https://mathworld.wolfram.com/IntegralDomain.html), also is a division ring (https://en.wikipedia.org/wiki/Division_ring), also has no zero divisors (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) other than 0 (for the special case that *n* = 1, it is the zero ring (https://en.wikipedia.org/wiki/Zero_ring, https://mathworld.wolfram.com/TrivialRing.html))). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid) and http://www.numericana.com/answer/primes.htm (the set of the primes) and http://www.numericana.com/answer/factoring.htm (factoring into primes). Besides, "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" to "the set of the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" to "the partially ordered binary relation by subsequence" is "the set of the prime numbers" to "the set of the integers > 1" to "the partially ordered binary relation by divisibility" (and indeed, the "> 1" in "the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" can be corresponded to the "> 1" in "the integers > 1") (for the reason why *b* itself is excluded (when *b* is prime, if *b* is composite, then there is no difference to include the *b* itself or not), see the sections below and https://mersenneforum.org/showpost.php?p=531632&postcount=7, the main reason is that *b* is the *only* prime ending with 0), thus the problem in this project is very important and beautiful.

|subsequence ordering|divisibility ordering|
|---|---|
|the "prime numbers > *b*" digit strings" in base *b*|the integers > 1|
|the set of the minimal elements of the "prime numbers > *b*" digit strings" in base *b* under the subsequence ordering (which is exactly the target of this project)|the set of the minimal elements of the integers > 1 under the divisibility ordering (which is exactly the set of the prime numbers)|
|no common subsequence with length > 1|coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime)|
|proper subsequence with length > 1|proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1|
|longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd)|
|shortest common supersequence (https://en.wikipedia.org/wiki/Shortest_common_supersequence)|least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm)|
|pairwise incomparable strings (no string is a subsequence of another string)|pairwise incomparable numbers (no number divides another number)|

This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf), https://github.com/curtisbright/mepn-data, https://github.com/curtisbright/mepn, https://github.com/RaymondDevillers/primes) to cover Conjectures ‘R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://t5k.org/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)) with *k*-values < *b*, i.e. finding the smallest prime of the form *k*×*b*<sup>*n*</sup>+1 and *k*×*b*<sup>*n*</sup>−1 (or proving that such prime does not exist) for all *k* < *b* (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showpost.php?p=144991&postcount=1, https://mersenneforum.org/showthread.php?t=10761, https://mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base *b* with *k*-values < *b*, i.e. finding the smallest prime of the form *b*<sup>*n*</sup>+*k* and *b*<sup>*n*</sup>−*k* (which are the dual forms of *k*×*b*<sup>*n*</sup>+1 and *k*×*b*<sup>*n*</sup>−1, respectively) (or proving that such prime does not exist) for all *k* < *b*) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as *b*<sup>*n*</sup>+2, *b*<sup>*n*</sup>−2, *b*<sup>*n*</sup>+(*b*−1), *b*<sup>*n*</sup>−(*b*−1), 2×*b*<sup>*n*</sup>+1, 2×*b*<sup>*n*</sup>−1, (*b*−1)×*b*<sup>*n*</sup>+1, (*b*−1)×*b*<sup>*n*</sup>−1, with *n* ≥ 1, for the same base *b* (of course, for some bases *b* the original minimal prime base *b* problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*<sup>*n*</sup>+1 if and only if *b*−1 is not prime, and the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*<sup>*n*</sup>−1 if and only if neither *b*−1 nor *b*−2 is prime, but I want the problem covers finding the smallest prime of these forms for *all* bases *b*)). The original minimal prime base *b* problem does not cover Conjectures ‘R Us Sierpinski/Riesel conjectures base *b* with conjectured *k* (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt) < *b*, since in Riesel side, the prime is not minimal prime in original definition if either *k*−1 or *b*−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if *k* is prime (e.g. 25×30<sup>34205</sup>−1 is not minimal prime in base 30 in original definition, since it is OT<sub>34205</sub> in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base *b* problem does. (There is someone else who also exclude the single-digit primes, but his research is about substring (https://en.wikipedia.org/wiki/Substring) instead of subsequence, see https://www.mersenneforum.org/showpost.php?p=235383&postcount=42, subsequences can contain consecutive elements which were not consecutive in the original sequence, a subsequence which consists of a consecutive run of elements from the original sequence, such as 234 from 123456, is a substring, substring is a refinement of the subsequence, subsequence is a generalization of substring, substring must be subsequence, but subsequence may not be substring, 514 is a subsequence of 352148, but not a substring of 352148, see the list below of the comparation of "subsequence" and "substring")

|subsequence|substring|
|---|---|
|https://oeis.org/A071062|https://oeis.org/A033274|
|https://oeis.org/A130448|https://oeis.org/A238334|
|https://oeis.org/A039995|https://oeis.org/A039997|
|https://oeis.org/A039994|https://oeis.org/A039996|
|https://oeis.org/A094535|https://oeis.org/A093301|
|https://oeis.org/A350508|https://oeis.org/A038103|
|https://oeis.org/A354113|https://oeis.org/A354114|
|longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|longest common substring (https://en.wikipedia.org/wiki/Longest_common_substring)

(we do not research substring in this project, because for substring ordering, "the set of the minimal elements of the base 10 representations of the prime numbers > 10" are very likely to be infinite (and thus almost unsolvable), since all primes of the form 1{0}3 (i.e. https://oeis.org/A159352, for the exponents see https://oeis.org/A049054, also see https://stdkmd.net/nrr/1/10003.htm) or 3{0}1 (i.e. https://oeis.org/A259866, for the exponents see https://oeis.org/A056807, also see https://stdkmd.net/nrr/3/30001.htm) are minimal elements of the base 10 representations of the prime numbers > 10 under the substring ordering, and there is likely infinitely many primes of the form 1{0}3 and infinitely many primes of the form 3{0}1, see https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes and https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) and http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) and https://mersenneforum.org/showpost.php?p=564786&postcount=3 and https://mersenneforum.org/showpost.php?p=461665&postcount=7 and https://mersenneforum.org/showpost.php?p=625978&postcount=1027)

However, including the base (*b*) itself results in automatic elimination of all possible extension numbers with "0 after 1" from the set (when the base is prime, if the base is composite, then there is no difference to include the base (*b*) itself or not), which is quite restrictive (since when the base is prime, then the base (*b*) itself is the only prime ending with 0, i.e. having trailing zero (https://en.wikipedia.org/wiki/Trailing_zero), since in any base, all numbers ending with 0 (i.e. having trailing zero) are divisible by the base (*b*), thus cannot be prime unless it is equal the base (*b*), i.e. "10" in base *b*, note that the numbers cannot have leading zero (https://en.wikipedia.org/wiki/Leading_zero), since typically this is not the way we write numbers (in any base), thus for all primes in our sets (i.e. all primes > base (*b*)), all zero digits must be "between" other digits). (for the reference of this, see https://mersenneforum.org/showpost.php?p=531632&postcount=7)

Besides, this problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not, i.e. no matter 1 is considered as prime or not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one), the sets in this problem are the same, while the sets in the original minimal prime problem are different, e.g. in base 10, if 1 is considered as prime, then the set in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set in this problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime.

The third reason for excluding the single-digit primes is that they are trivial like that Conjectures ‘R Us Sierpinski/Riesel conjectures base *b* requires exponent *n* ≥ 1 for these primes (see https://mersenneforum.org/showpost.php?p=447679&postcount=27), *n* = 0 is not acceptable to avoid the trivial primes (e.g. 2×*b*<sup>*n*</sup>+1, 4×*b*<sup>*n*</sup>+1, 6×*b*<sup>*n*</sup>+1, 10×*b*<sup>*n*</sup>+1, 12×*b*<sup>*n*</sup>+1, 16×*b*<sup>*n*</sup>+1, 3×*b*<sup>*n*</sup>−1, 4×*b*<sup>*n*</sup>−1, 6×*b*<sup>*n*</sup>−1, 8×*b*<sup>*n*</sup>−1, 12×*b*<sup>*n*</sup>−1, 14×*b*<sup>*n*</sup>−1, ... cannot be quickly eliminated with *n* = 0, or the conjectures become much more easy and uninteresting), for the same reason, this minimal prime puzzle requires ≥ *b* (i.e. ≥ 2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. families containing digit 2, 3, 5, 7, B, D, H, J, N, T, V, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much more easy and uninteresting).

The fourth reason for excluding the primes ≤ *b* is that starting with *b*+1 makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base *b* more simple and smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683), it is (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955), where *eulerphi* is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, https://oeis.org/A000010), since *b*−1 is the number of possible first digit (except 0, all digits can be first digit), and *eulerphi*(*b*) is the number of possible last digit (only digits coprime to *b* can be last digit), by rule of product, there are (*b*−1)×*eulerphi*(*b*) possible (first digit,last digit) combo, and if start with *b*, then when *b* is prime, there is an additional possible (first digit,last digit) combo: (1,0), and hence the formula will be (*b*−1)×*eulerphi*(*b*)+1 if *b* is prime, or (*b*−1)×*eulerphi*(*b*) if *b* is composite (the fully formula will be (*b*−1)×*eulerphi*(*b*)+*isprime*(*b*) or (*b*−1)×*eulerphi*(*b*)+*floor*((*b*−*eulerphi*(*b*)) / (*b*−1))), which is more complex, and if start with 1 (i.e. the original minimal prime problem), the formula is much more complex.

It is found that both "number of minimal primes base *b*" and "length of the largest minimal prime base *b*" are roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup>, where *e* = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), *γ* = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), *eulerphi* is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, https://oeis.org/A000010), you can see the condensed table for bases 2 ≤ *b* ≤ 36 in the bottom of this article, *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup> is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955), and since (*b*−1)×*eulerphi*(*b*) has polynomial growth (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) for *b* (since it is always between *b*−1 and *b*<sup>2</sup>), thus *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup> has exponential growth for *b*, and "largest minimal prime base *b*" is roughly *b*<sup>*e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup></sup>, which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for *b*. (there are also asymptotic analysis for other sets of primes in various bases *b*, such as the left-truncatable primes and the right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/) in various bases *b*, see http://chesswanks.com/num/LTPs/ for the left-truncatable primes in bases *b* ≤ 120 and http://fatphil.org/maths/rtp/rtp.html for the right-truncatable primes in bases *b* ≤ 90)

The lengths of the minimal primes in base *b* appear to follow the Zipf's law (https://en.wikipedia.org/wiki/Zipf%27s_law, https://mathworld.wolfram.com/ZipfsLaw.html) (which is similar to the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html)), for *n* ≥ 2, the number of *n*-digit minimal primes in base *b* is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to *n*−1 (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is *c*/(*n*−1), where c is a fixed constant) (i.e. the graph of the points (*x*,*y* = the number of *x*-digit minimal primes in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/(*x*−1) in the *xy*-plane for a fixed real number *c*), for any fixed base *b*, also, for *n* ≥ 1, the length of the *n*th largest minimal prime in base *b* is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to *n* (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is *c*/*n*, where c is a fixed constant) (i.e. the graph of the points (*x*,*y* = the length of the *x*th largest minimal prime in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/*x* in the *xy*-plane for a fixed real number *c*), for any fixed base *b*.

We can use the sense of http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) to say: (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A, this family has been searched to *n* = 500000 with no prime or probable prime found, we can use ">500000" for the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A (while for the *n* of the smallest prime in the base 13 family 95<sub>*n*</sub>, it is 197420), ">500000" includes infinity (since infinity is > 500000) but does not includes 0 or −1, it is still possible that there is no prime in the base 13 family A3<sub>*n*</sub>A, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set *S* of the numbers *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, then this set *S* is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest *n* ≥ 1 such that *k*×2<sup>*n*</sup>+1 is prime" is ∞ for *k* = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for *k* = 10223 and 13018586 for *k* = 19249, another example is "the smallest *n* such that (*b*<sup>*n*</sup>−1)/(*b*−1) is prime" is ∞ for *b* = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for *b* = 691 and 41189 for *b* = 693))

e.g. for bases *b* = 23 and *b* = 25:

* We have a 50% chance of solving the "minimal prime problem" at length 10<sup>25</sup>.
* We have a 5% chance of solving the "minimal prime problem" at length 10<sup>16</sup>.
* We have a 95% chance of solving the "minimal prime problem" at length 10<sup>48</sup>.
* The chances at lengths 10<sup>6</sup>, 10<sup>7</sup>, 10<sup>8</sup> are respectively 10<sup>−86</sup>, 10<sup>−52</sup>, and 10<sup>−33</sup>.

This problem covers finding the smallest prime in these families in the same base *b* (or proving that such prime does not exist), since the smallest prime in these families (if exists) **must** be a minimal prime in base *b*: (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) *b*)

|family|smallest allowed *b*|smallest allowed *n*|*OEIS* sequences for the smallest *n* such that this form is prime for fixed base *b* (such *n* always exist unless these families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them) if my conjecture is true)|*OEIS* sequences for the smallest base *b* such that this form is prime for fixed *n* (such base *b* always exist unless these families can be ruled out as only containing composites (by single prime factor or algebraic factorization) if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true)|references|current smallest base *b* such that this family is an unsolved family (i.e. have no known prime (or strong probable prime) members > *b*, nor can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them))|search limit of the length of this family in this base *b*|smaller bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them)<br>bases *b*: why this family contain no primes > *b*|smaller bases *b* with the smallest (probable) prime in this family has length > 100: *b* (*length*)|
|---|---|---|---|---|---|---|---|---|---|
|(*b*<sup>*n*</sup>−1)/(*b*−1)|2|2|https://oeis.org/A084740<br>https://oeis.org/A084738 (corresponding primes)<br>https://oeis.org/A246005 (odd *b*)<br>https://oeis.org/A065854 (prime *b*)<br>https://oeis.org/A279068 (prime *b*, corresponding primes)<br>https://oeis.org/A360738 (*n* replaced by *n*−1)<br>https://oeis.org/A279069 (prime *b*, *n* replaced by *n*−1)<br>https://oeis.org/A065813 (prime *b*, *n* replaced by (*n*−1)/2)<br>https://oeis.org/A128164 (*n* = 2 not allowed)<br>https://oeis.org/A285642 (*n* = 2 not allowed, corresponding primes)|https://oeis.org/A066180<br>https://oeis.org/A084732 (corresponding primes)<br>**(if this form is prime, then *n* must be a prime, see https://t5k.org/notes/proofs/Theorem2.html for the proof, this proof can be generalized to any base *b*)**|http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German)<br>https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html<br>http://www.primenumbers.net/Henri/us/MersFermus.htm<br>http://www.bitman.name/math/table/379 (in Italian)<br>https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf)<br>https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)|185|100000|*b* = *m*<sup>2</sup>: difference-of-two-squares factorization<br>*b* = *m*<sup>3</sup>: difference-of-two-cubes factorization<br>*b* = *m*<sup>5</sup>: difference-of-two-5th-powers factorization<br>*b* = *m*<sup>7</sup>: difference-of-two-7th-powers factorization<br>**(note: although bases *b* = 4, 8, 16, 27, 36, 100, 128 have algebraic factorization (if the numbers are factored as *F* × *G* / *d*), *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial, thus these bases *b* have only one very small prime > *b* instead of "can be ruled out as only containing composites (only count the numbers > *b*)", thus the only smaller bases *b* such that this family can be ruled out as only containing composites (only count the numbers > *b*) are 9, 25, 32, 49, 64, 81, 121, 125, 144, 169)**|35 (313)<br>39 (349)<br>47 (127)<br>51 (4229)<br>91 (4421)<br>92 (439)<br>124 (599)<br>135 (1171)<br>139 (163)<br>142 (1231)<br>152 (270217)<br>171 (181)<br>174 (3251)<br>182 (167)<br>183 (223)<br>184 (16703)|
|*b*<sup>*n*</sup>+1|2|1|https://oeis.org/A079706<br>https://oeis.org/A084712 (corresponding primes)<br>https://oeis.org/A228101 (*log*<sub>2</sub> of *n*)<br>https://oeis.org/A123669 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A056993<br>https://oeis.org/A123599 (corresponding primes)<br>**(if this form is prime, then *n* must be a power of 2, see http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*)**|http://jeppesn.dk/generalized-fermat.html<br>http://www.noprimeleftbehind.net/crus/GFN-primes.htm<br>http://yves.gallot.pagesperso-orange.fr/primes/index.html<br>http://yves.gallot.pagesperso-orange.fr/primes/results.html<br>http://yves.gallot.pagesperso-orange.fr/primes/stat.html|38|33554432|*b* == 1 mod 2: always divisible by 2<br>*b* = *m*<sup>3</sup>: sum-of-two-cubes factorization<br>*b* = *m*<sup>5</sup>: sum-of-two-5th-powers factorization|(none)|
|(*b*<sup>*n*</sup>+1)/2|3<br>(only odd *b*)|2||https://oeis.org/A275530<br>**(if this form is prime, then *n* must be a power of 2, see http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base *b*)**|http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German)|31|16777215|*b* = *m*<sup>3</sup>: sum-of-two-cubes factorization|(none)|
|2×*b*<sup>*n*</sup>+1|3|1|https://oeis.org/A119624<br>https://oeis.org/A253178 (only bases *b* which have possible primes)<br>https://oeis.org/A098872 (*b* divisible by 6)||https://mersenneforum.org/showthread.php?t=6918<br>https://mersenneforum.org/showthread.php?t=19725 (*b* == 11 mod 12)<br>https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|365|500000|*b* == 1 mod 3: always divisible by 3|38 (2730)<br>47 (176)<br>101 (192276)<br>104 (1234)<br>117 (287)<br>122 (756)<br>137 (328)<br>147 (155)<br>167 (6548)<br>203 (106)<br>206 (46206)<br>218 (333926)<br>236 (161230)<br>248 (322)<br>257 (12184)<br>263 (958)<br>287 (5468)<br>305 (16808)<br>347 (124)<br>353 (2314)|
|2×*b*<sup>*n*</sup>−1|3|1|https://oeis.org/A119591<br>https://oeis.org/A098873 (*b* divisible by 6)|https://oeis.org/A157922|https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217<br>https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|581|600000|(none)|29 (137)<br>67 (769)<br>74 (133)<br>107 (21911)<br>152 (797)<br>161 (229)<br>170 (166429)<br>191 (971)<br>215 (1073)<br>224 (109)<br>233 (8621)<br>235 (181)<br>254 (2867)<br>260 (121)<br>276 (2485)<br>278 (43909)<br>284 (417)<br>298 (4203)<br>303 (40175)<br>308 (991)<br>347 (523)<br>380 (3787)<br>382 (2325)<br>383 (20957)<br>393 (108)<br>395 (397)<br>401 (113)<br>418 (472)<br>422 (541)<br>431 (529)<br>434 (1167)<br>449 (175)<br>457 (103)<br>473 (661)<br>480 (145)<br>503 (861)<br>513 (299)<br>515 (58467)<br>522 (62289)<br>524 (165)<br>536 (841)<br>550 (1381)<br>551 (2719)<br>572 (3805)<br>578 (129469)|
|*b*<sup>*n*</sup>+2|3|1|https://oeis.org/A138066<br>https://oeis.org/A084713 (corresponding primes)<br>https://oeis.org/A138067 (*n* = 1 not allowed)|https://oeis.org/A087576<br>https://oeis.org/A095302 (corresponding primes)||167|100000|*b* == 0 mod 2: always divisible by 2<br>*b* == 1 mod 3: always divisible by 3|47 (114)<br>89 (256)<br>159 (137)|
|*b*<sup>*n*</sup>−2|3|2|https://oeis.org/A250200<br>https://oeis.org/A255707 (*n* = 1 allowed)<br>https://oeis.org/A084714 (*n* = 1 allowed, corresponding primes)<br>https://oeis.org/A292201 (prime *b*, *n* = 1 allowed)|https://oeis.org/A095303<br>https://oeis.org/A095304 (corresponding primes)|https://www.primepuzzles.net/puzzles/puzz_887.htm (*n* = 1 allowed)|305|30000|*b* == 0 mod 2: always divisible by 2|81 (130)<br>97 (747)<br>197 (164)<br>209 (126)<br>215 (134)<br>221 (552)<br>287 (3410)|
|3×*b*<sup>*n*</sup>+1|4|1|https://oeis.org/A098877 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|718|300000|*b* == 1 mod 2: always divisible by 2|108 (271)<br>314 (281)<br>358 (9561)<br>386 (184)<br>424 (1106)<br>458 (108)<br>492 (157)<br>636 (142)<br>646 (159)<br>648 (647)<br>652 (621)<br>654 (217)<br>690 (358)|
|3×*b*<sup>*n*</sup>−1|4|1|https://oeis.org/A098876 (*b* divisible by 6)||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|588|500000|*b* == 1 mod 2: always divisible by 2|42 (2524)<br>202 (263)<br>212 (283)<br>238 (105)<br>422 (191)<br>432 (16003)<br>446 (4851)<br>452 (335)<br>464 (219)<br>522 (347)<br>532 (136)<br>572 (377)<br>582 (445)|
|4×*b*<sup>*n*</sup>+1|5|1||**(such base *b* does not exist if *n* is divisible by 4)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|32|1717986918|*b* == 1 mod 5: always divisible by 5<br>*b* == 14 mod 15: always divisible by some element of {3,5}|23 (343)|
|4×*b*<sup>*n*</sup>−1|5|1||**(such base *b* does not exist if *n* is even)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|275|600000|*b* == 1 mod 3: always divisible by 3<br>*b* == 14 mod 15: always divisible by some element of {3,5}<br>*b* = *m*<sup>2</sup>: difference-of-two-squares factorization<br>*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|47 (1556)<br>72 (1119850)<br>107 (252)<br>167 (1866)<br>212 (34414)<br>218 (23050)<br>236 (940)<br>240 (1402)<br>251 (272)<br>261 (820)<br>270 (89662)|
|5×*b*<sup>*n*</sup>+1|6|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|308|300000|*b* == 1 mod 2: always divisible by 2<br>*b* == 1 mod 3: always divisible by 3|122 (136)<br>170 (176)<br>200 (768)<br>248 (262)<br>266 (510)|
|5×*b*<sup>*n*</sup>−1|6|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|338|300000|*b* == 1 mod 2: always divisible by 2|14 (19699)<br>68 (13575)<br>112 (133)<br>116 (157)<br>196 (9850)<br>206 (109)<br>254 (15451)<br>320 (233)|
|6×*b*<sup>*n*</sup>+1|7|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|212|500000|*b* == 1 mod 7: always divisible by 7<br>*b* == 34 mod 35: always divisible by some element of {5,7}|53 (144)<br>67 (4533)<br>93 (521)<br>108 (16318)<br>129 (16797)<br>144 (783)<br>163 (1304)<br>185 (171)<br>193 (149)|
|6×*b*<sup>*n*</sup>−1|7|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|234|600000|*b* == 1 mod 5: always divisible by 5<br>*b* == 34 mod 35: always divisible by some element of {5,7}<br>*b* = 6×*m*<sup>2</sup> with *m* == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization|48 (295)<br>118 (211)<br>119 (666)<br>154 (1990)<br>178 (119)<br>188 (951)|
|7×*b*<sup>*n*</sup>+1|8|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|1136|10000|*b* == 1 mod 2: always divisible by 2|50 (517)<br>62 (309)<br>170 (179)<br>194 (281)<br>224 (689)<br>236 (347)<br>308 (107)<br>338 (793)<br>380 (475)<br>382 (519)<br>386 (121)<br>398 (17473)<br>434 (321)<br>466 (181)<br>500 (1997)<br>520 (198)<br>522 (235)<br>524 (127)<br>598 (423)<br>632 (8447)<br>638 (265)<br>644 (3379)<br>652 (185)<br>674 (181)<br>682 (796)<br>724 (388)<br>734 (189)<br>764 (189)<br>836 (5701)<br>868 (274)<br>892 (157)<br>920 (491)<br>926 (523)<br>930 (218)<br>958 (169)<br>960 (128)<br>974 (1589)<br>982 (313)<br>1004 (54849)<br>1082 (2113)<br>1102 (820)|
|7×*b*<sup>*n*</sup>−1|8|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|308|300000|*b* == 1 mod 2: always divisible by 2<br>*b* == 1 mod 3: always divisible by 3|68 (25396)<br>182 (210)<br>198 (117)<br>248 (3180)<br>260 (826)|
|8×*b*<sup>*n*</sup>+1|9|1||**(such base *b* does not exist if *n* is divisible by 3)**|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|86|1000000|*b* == 1 mod 3: always divisible by 3<br>*b* == 20 mod 21: always divisible by some element of {3,7}<br>*b* == 47, 83 mod 195: always divisible by some element of {3,5,13}<br>*b* = *m*<sup>3</sup>: sum-of-two-cubes factorization|23 (119216)<br>53 (227184)<br>68 (320)|
|8×*b*<sup>*n*</sup>−1|9|1||**(such base *b* does not exist if *n* is divisible by 3)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|321|600000|*b* == 1 mod 7: always divisible by 7<br>*b* == 20 mod 21: always divisible by some element of {3,7}<br>*b* == 83, 307 mod 455: always divisible by some element of {5,7,13}<br>*b* = *m*<sup>3</sup>: difference-of-two-cubes factorization|97 (192336)<br>101 (113)<br>112 (269)<br>131 (197)<br>145 (6369)<br>170 (15423)<br>194 (38361)<br>202 (155772)<br>217 (179)<br>237 (528)<br>245 (501)<br>252 (6288)<br>270 (108)<br>277 (1229)<br>282 (21413)<br>283 (164769)<br>284 (5267)|
|9×*b*<sup>*n*</sup>+1|10|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|724|500000|*b* == 1 mod 2: always divisible by 2<br>*b* == 1 mod 5: always divisible by 5|94 (264)<br>134 (184)<br>182 (264)<br>244 (1836)<br>248 (39511)<br>332 (311)<br>334 (340)<br>344 (306)<br>364 (166)<br>400 (265)<br>402 (127)<br>422 (106)<br>448 (372)<br>454 (136)<br>490 (469)<br>534 (106)<br>544 (4706)<br>592 (96870)<br>622 (127)<br>634 (190)<br>664 (290)|
|9×*b*<sup>*n*</sup>−1|10|1||**(such base *b* does not exist if *n* is even)**|https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|378|300000|*b* == 1 mod 2: always divisible by 2<br>*b* = *m*<sup>2</sup>: difference-of-two-squares factorization<br>*b* == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization|88 (172)<br>112 (5718)<br>116 (250)<br>130 (468)<br>138 (35686)<br>188 (3888)<br>198 (304)<br>218 (178)<br>258 (106)<br>286 (164)<br>292 (2928)<br>328 (606)<br>332 (946)<br>346 (130)<br>360 (316)<br>366 (238)|
|10×*b*<sup>*n*</sup>+1|11|1|https://oeis.org/A088782<br>https://oeis.org/A088622 (corresponding primes)|https://oeis.org/A089319<br>https://oeis.org/A089318 (corresponding primes)|https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|185|1000000|*b* == 1 mod 11: always divisible by 11<br>*b* == 32 mod 33: always divisible by some element of {3,11}|17 (1357)<br>61 (166)<br>74 (139)<br>101 (1507)<br>137 (103)<br>142 (408)<br>173 (264235)<br>176 (147)<br>179 (337)|
|10×*b*<sup>*n*</sup>−1|11|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|233|600000|*b* == 1 mod 3: always divisible by 3<br>*b* == 32 mod 33: always divisible by some element of {3,11}|17 (118)<br>80 (423716)<br>89 (250)<br>185 (6784)<br>194 (3150)<br>215 (144)|
|11×*b*<sup>*n*</sup>+1|12|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|560|100000|*b* == 1 mod 2: always divisible by 2<br>*b* == 1 mod 3: always divisible by 3<br>*b* == 14 mod 15: always divisible by some element of {3,5}|68 (3948)<br>108 (190)<br>110 (162)<br>152 (838)<br>222 (101)<br>236 (154)<br>294 (365)<br>320 (1264)<br>384 (491)<br>392 (412)<br>432 (226)<br>440 (146)<br>462 (762)<br>506 (270)<br>528 (249)<br>534 (689)<br>542 (4910)|
|11×*b*<sup>*n*</sup>−1|12|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|214|1000000|*b* == 1 mod 2: always divisible by 2<br>*b* == 1 mod 5: always divisible by 5<br>*b* == 14 mod 15: always divisible by some element of {3,5}|38 (767)<br>68 (199)<br>72 (2446)<br>80 (209)<br>102 (2071)<br>140 (109)<br>170 (109)<br>178 (178)<br>188 (183)|
|12×*b*<sup>*n*</sup>+1|13|1|||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|163|500000|*b* == 1 mod 13: always divisible by 13<br>*b* == 142 mod 143: always divisible by some element of {11,13}|30 (1024)<br>65 (685)<br>67 (136)<br>68 (656922)<br>82 (108)<br>87 (1215)<br>102 (2740)<br>106 (139)<br>159 (122)|
|12×*b*<sup>*n*</sup>−1|13|1|||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n<br>https://mersenneforum.org/showthread.php?t=10354|263|314000|*b* == 1 mod 11: always divisible by 11<br>*b* == 142 mod 143: always divisible by some element of {11,13}|43 (204)<br>65 (1194)<br>98 (3600)<br>129 (229)<br>147 (113)<br>153 (21660)<br>186 (112718)<br>193 (117)<br>230 (188)|
|(*b*−1)×*b*<sup>*n*</sup>+1|2|1|https://oeis.org/A305531<br>https://oeis.org/A087139 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 1 mod 6 except *n* = 1)**|https://www.rieselprime.de/ziki/Williams_prime_MP_least<br>https://www.rieselprime.de/ziki/Williams_prime_MP_table<br>https://sites.google.com/view/williams-primes<br>http://www.bitman.name/math/table/477 (in Italian)|123|400000|(none)|53 (961)<br>65 (947)<br>77 (829)<br>88 (3023)<br>122 (6217)|
|(*b*−1)×*b*<sup>*n*</sup>−1|2|1|https://oeis.org/A122396 (prime *b*, *n* replaced by *n*+1)|**(such base *b* does not exist if *n* == 4 mod 6)**|https://harvey563.tripod.com/wills.txt<br>https://www.rieselprime.de/ziki/Williams_prime_MM_least<br>https://www.rieselprime.de/ziki/Williams_prime_MM_table<br>https://sites.google.com/view/williams-primes<br>http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf)<br>https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf)<br>http://www.bitman.name/math/table/484 (in Italian)|128|2450000|(none)|26 (134)<br>38 (136212)<br>62 (900)<br>83 (21496)<br>91 (520)<br>93 (477)<br>98 (4984)<br>108 (411)<br>113 (286644)<br>125 (8740)|
|*b*<sup>*n*</sup>+(*b*−1)|2|1|https://oeis.org/A076845<br>https://oeis.org/A076846 (corresponding primes)<br>https://oeis.org/A078178 (*n* = 1 not allowed)<br>https://oeis.org/A078179 (*n* = 1 not allowed, corresponding primes)|https://oeis.org/A248079<br>**(such base *b* does not exist if *n* == 5 mod 6)**|https://sites.google.com/view/williams-primes|257|12000|(none)|32 (109)<br>80 (195)<br>107 (1401)<br>113 (20089)<br>123 (64371)<br>128 (505)<br>161 (105)<br>173 (11429)<br>179 (3357)<br>197 (977)<br>212 (109)<br>224 (259)<br>227 (157)<br>237 (110)<br>238 (117)|
|*b*<sup>*n*</sup>−(*b*−1)|2|2|https://oeis.org/A113516<br>https://oeis.org/A343589 (corresponding primes)|https://oeis.org/A113517<br>**(such base *b* does not exist if *n* == 2 mod 6 except *n* = 2)**|https://sites.google.com/view/williams-primes<br>https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime *b*)<br>http://www.bitman.name/math/table/435 (in Italian) (prime *b*)|93|60000|(none)|71 (3019)<br>82 (169)<br>83 (965)<br>88 (2848)|

(below (as well as the "left *b*" files), family "12{3}45" means sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...}, where the members are expressed as base *b* strings, like the numbers in https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/abbbc.htm, https://stdkmd.net/nrr/prime/primesize.txt, https://stdkmd.net/nrr/prime/primesize.zip, https://stdkmd.net/nrr/prime/primecount.htm, https://stdkmd.net/nrr/prime/primecount.txt, https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, e.g. 1{3} (in decimal) is the numbers in https://stdkmd.net/nrr/1/13333.htm, and {1}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11113.htm, and 1{2}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/12223.htm, also, superscripts always means exponents (https://en.wikipedia.org/wiki/Exponentiation, https://www.rieselprime.de/ziki/Exponent, https://mathworld.wolfram.com/Exponent.html, https://mathworld.wolfram.com/Power.html, https://mathworld.wolfram.com/Exponentiation.html), subscripts are always used to indicate repetitions of digits, e.g. 123<sub>4</sub>567 = 123333567, all subscripts are written in decimal)

In fact, this problem covers finding the smallest prime of these form in the same base *b*: (where *x*, *y*, *z* are any digits in base *b*)

* *x*{0}*y*
* *x*{*y*} (unless *y* = 1) (see https://stdkmd.net/nrr/abbbb.htm)
* {*x*}*y* (unless *x* = 1) (see https://stdkmd.net/nrr/aaaab.htm)
* *x*{0}*yz* (unless there is a prime of the form *x*{0}*y* or *x*{0}*z*)
* *xy*{0}*z* (unless there is a prime of the form *x*{0}*z* or *y*{0}*z*)
* *xy*{*x*} (unless either *x* = 1 or there is a prime of the form *y*{*x*} (or both)) (see https://stdkmd.net/nrr/abaaa.htm)
* {*x*}*yx* (unless either *x* = 1 or there is a prime of the form {*x*}*y* (or both)) (see https://stdkmd.net/nrr/aaaba.htm)

The primes in forms *x*{*y*}, {*x*}*y*, *xy*{*x*}, {*x*}*yx* in base *b* are near-repdigit primes (https://t5k.org/glossary/xpage/NearRepdigitPrime.html, https://t5k.org/top20/page.php?id=15, https://pzktupel.de/Primetables/TableNRD.php, https://oeis.org/A164937, https://stdkmd.net/nrr/#factortables_nr, https://stdkmd.net/nrr/#factortables_np, https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/records.htm#nrprime, https://stdkmd.net/nrr/records.htm#nrprp, https://stdkmd.net/nrr/records.htm#nrpprime, https://stdkmd.net/nrr/records.htm#nrpprp, http://factordb.com/tables.php?open=1, http://factordb.com/tables.php?open=3) in base *b*.

Proving that "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = the set *S* is equivalent to (https://en.wikipedia.org/wiki/Logical_equivalence):

* Prove that all elements in *S*, when read as base *b* representation, are primes > *b*.
* Prove that all proper subsequence of all elements in *S*, when read as base *b* representation, which are > *b*, are composite (https://en.wikipedia.org/wiki/Composite_number, https://t5k.org/glossary/xpage/Composite.html, https://www.rieselprime.de/ziki/Composite_number, https://mathworld.wolfram.com/CompositeNumber.html, https://oeis.org/A002808).
* Prove that all primes > *b*, when written in base *b*, contain at least one element in *S* as subsequence (equivalently, prove that all strings not containing any element in *S* as subsequence, when read as base *b* representation, which are > *b*, are composite).

("the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = *S* is proved if and only if all these three problems are proved, i.e. "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" = *S* is a theorem if and only if all these three "conjectures" are theorems)

e.g. proving that "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10.
* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite.
* Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite, since they are contraposition (https://en.wikipedia.org/wiki/Contraposition), *P* ⟶ *Q* and ¬*Q* ⟶ ¬*P* are logically equivalent (https://en.wikipedia.org/wiki/Logical_equivalence)).

(since for base *b* = 10, all these three problems are proved, i.e. all they are theorems, thus, "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also proved, i.e. "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also a theorem)

Theorem (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/): The set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/): (this proof uses the notation in http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf), i.e. "*X* ◁ *Y*" means "*X* is a subsequence of *Y*") (below, *𝜆* is the empty string (https://en.wikipedia.org/wiki/Empty_string)) (**bold** for minimal primes)

Assume *p* is a prime > 10, and the last digit of *p* must lie in {1,3,7,9}

Case 1: *p* ends with 1.

In this case we can write *p* = *x*1. If *x* contains 1, 3, 4, 6, or 7, then (respectively) **11** ◁ *p*, **31** ◁ *p*, **41** ◁ *p*, **61** ◁ *p*, or **71** ◁ *p*. Hence we may assume all digits of *x* are 0, 2, 5, 8, or 9.

Case 1.1: *p* begins with 2.

In this case we can write *p* = 2*y*1. If 5 ◁ *y*, then **251** ◁ *p*. If 8 ◁ *y*, then **281** ◁ *p*. If 9 ◁ *y*, then 29 ◁ *p*. Hence we may assume all digits of *y* are 0 or 2.

If 22 ◁ *y*, then **2221** ◁ *p*. Hence we may assume *y* contains zero or one 2's.

If *y* contains no 2's, then *p* ∈ 2{0}1. But then, since the sum of the digits of *p* is 3, *p* is divisible by 3, so *p* cannot be prime.

If *y* contains exactly one 2, then we can write *p* = 2*z*2*w*1, where *z*,*w* ∈ {0}. If 0 ◁ *z* and 0 ◁ *w*, then **20201** ◁ *p*. Hence we may assume either *z* or *w* is empty.

If *z* is empty, then *p* ∈ 22{0}1, and the smallest prime *p* ∈ 22{0}1 is **22000001**.

If *w* is empty, then *p* ∈ 2{0}21, and the smallest prime *p* ∈ 2{0}21 is **20021**.

Case 1.2: *p* begins with 5.

In this case we can write *p* = 5*y*1. If 2 ◁ *y*, then **521** ◁ *p*. If 9 ◁ *y*, then 59 ◁ *p*. Hence we may assume all digits of *y* are 0, 5, or 8.

If 05 ◁ *y*, then **5051** ◁ *p*. If 08 ◁ *y*, then **5081** ◁ *p*. If 50 ◁ *y*, then **5501** ◁ *p*. If 58 ◁ *y*, then **5581** ◁ *p*. If 80 ◁ *y*, then **5801** ◁ *p*. If 85 ◁ *y*, then **5851** ◁ *p*. Hence we may assume *y* ∈ {0} ∪ {5} ∪ {8}.

If *y* ∈ {0}, then *p* ∈ 5{0}1. But then, since the sum of the digits of *p* is 6, *p* is divisible by 3, so *p* cannot be prime.

If *y* ∈ {5}, then *p* ∈ 5{5}1, and the smallest prime *p* ∈ 5{5}1 is **555555555551**.

If *y* ∈ {8}, since if 88 ◁ *y*, then 881 ◁ *p*, hence we may assume *y* ∈ {*𝜆*, 8}, and thus *p* ∈ {51, 581}, but 51 and 581 are both composite.

Case 1.3: *p* begins with 8.

In this case we can write p = 8*y*1. If 2 ◁ *y*, then **821** ◁ *p*. If 8 ◁ *y*, then **881** ◁ *p*. If 9 ◁ *y*, then 89 ◁ *p*. Hence we may assume all digits of *y* are 0 or 5.

If 50 ◁ *y*, then **8501** ◁ *p*. Hence we may assume y ∈ {0}{5}.

If 005 ◁ *y*, then **80051** ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If *y* ∈ {0}, then *p* ∈ 8{0}1. But then, since the sum of the digits of *p* is 9, *p* is divisible by 3, so *p* cannot be prime.

If *y* ∈ {5}, since if 55555555555 ◁ *y*, then 555555555551 ◁ *p*, hence we may assume *y* ∈ {*𝜆*, 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus *p* ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If *y* ∈ 0{5}, since if 55555555555 ◁ *y*, then 555555555551 ◁ *p*, hence we may assume *y* ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus *p* ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only **80555551** is minimal prime.

Case 1.4: *p* begins with 9.

In this case we can write p = 9*y*1. If 9 ◁ *y*, then **991** ◁ *p*. Hence we may assume all digits of *y* are 0, 2, 5, or 8.

If 00 ◁ *y*, then **9001** ◁ *p*. If 22 ◁ *y*, then **9221** ◁ *p*. If 55 ◁ *y*, then **9551** ◁ *p*. If 88 ◁ *y*, then 881 ◁ *p*. Hence we may assume *y* contains at most one 0, at most one 2, at most one 5, and at most one 8.

If *y* only contains at most one 0 and does not contain any of {2, 5, 8}, then *y* ∈ {*𝜆*, 0}, and thus *p* ∈ {91, 901}, but 91 and 901 are both composite. If *y* only contains at most one 0 and only one of {2, 5, 8}, then the sum of the digits of *p* is divisible by 3, *p* is divisible by 3, so *p* cannot be prime. Hence we may assume *y* contains at least two of {2, 5, 8}.

If 25 ◁ *y*, then 251 ◁ *p*. If 28 ◁ *y*, then 281 ◁ *p*. If 52 ◁ *y*, then 521 ◁ *p*. If 82 ◁ *y*, then 821 ◁ *p*. Hence we may assume *y* contains no 2's (since if *y* contains 2, then *y* cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ *y*, then **9851** ◁ *p*. Hence we may assume *y* ∈ {58, 580, 508, 058}, and thus *p* ∈ {9581, 95801, 95081, 90581}, and of these numbers only 95801 is prime, but 95801 is not minimal prime since 5801 ◁ 95801.

Case 2: *p* ends with 3.

In this case we can write p = *x*3. If *x* contains 1, 2, 4, 5, 7, or 8, then (respectively) **13** ◁ *p*, **23** ◁ *p*, **43** ◁ *p*, **53** ◁ *p*, **73** ◁ *p*, or **83** ◁ *p*. Hence we may assume all digits of *x* are 0, 3, 6, or 9, and thus all digits of *p* are 0, 3, 6, or 9. But then, since the digits of *p* all have a common factor 3, *p* is divisible by 3, so *p* cannot be prime.

Case 3: *p* ends with 7.

In this case we can write *p* = *x*7. If *x* contains 1, 3, 4, 6, or 9, then (respectively) **17** ◁ *p*, **37** ◁ *p*, **47** ◁ *p*, **67** ◁ *p*, or **97** ◁ *p*. Hence we may assume all digits of *x* are 0, 2, 5, 7, or 8.

Case 3.1: *p* begins with 2.

In this case we can write *p* = 2*y*7. If 2 ◁ *y*, then **227** ◁ *p*. If 5 ◁ *y*, then **257** ◁ *p*. If 7 ◁ *y*, then **277** ◁ *p*. Hence we may assume all digits of *y* are 0 or 8.

If 08 ◁ *y*, then **2087** ◁ *p*. If 88 ◁ *y*, then 887 ◁ *p*. Hence we may assume *y* ∈ {0} ∪ 8{0}.

If *y* ∈ {0}, then *p* ∈ 2{0}7. But then, since the sum of the digits of *p* is 9, *p* is divisible by 3, so *p* cannot be prime.

If y ∈ 8{0}, then *p* ∈ 28{0}7. But then *p* is divisible by 7, since for *n* ≥ 0 we have 7 × 40<sub>*n*</sub>1 = 280<sub>*n*</sub>7.

Case 3.2: *p* begins with 5.

In this case we can write *p* = 5*y*7. If 5 ◁ *y*, then **557** ◁ *p*. If 7 ◁ *y*, then **577** ◁ *p*. If 8 ◁ *y*, then **587** ◁ *p*. Hence we may assume all digits of *y* are 0 or 2.

If 22 ◁ *y*, then 227 ◁ *p*. Hence we may assume *y* contains zero or one 2's.

If *y* contains no 2's, then *p* ∈ 5{0}7. But then, since the sum of the digits of *p* is 12, *p* is divisible by 3, so *p* cannot be prime.

If *y* contains exactly one 2, then we can write *p* = 5*z*2*w*7, where *z*,*w* ∈ {0}. If 0 ◁ *z* and 0 ◁ *w*, then **50207** ◁ *p*. Hence we may assume either *z* or *w* is empty.

If *z* is empty, then *p* ∈ 52{0}7, and the smallest prime *p* ∈ 52{0}7 is **5200007**.

If *w* is empty, then *p* ∈ 5{0}27, and the smallest prime *p* ∈ 5{0}27 is **5000000000000000000000000000027**.

Case 3.3: *p* begins with 7.

In this case we can write *p* = 7*y*7. If 2 ◁ *y*, then **727** ◁ *p*. If 5 ◁ *y*, then **757** ◁ *p*. If 8 ◁ *y*, then **787** ◁ *p*. Hence we may assume all digits of *y* are 0 or 7, and thus all digits of *p* are 0 or 7. But then, since the digits of *p* all have a common factor 7, *p* is divisible by 7, so *p* cannot be prime.

Case 3.4: *p* begins with 8.

In this case we can write *p* = 8*y*7. If 2 ◁ *y*, then **827** ◁ *p*. If 5 ◁ *y*, then **857** ◁ *p*. If 7 ◁ *y*, then **877** ◁ *p*. If 8 ◁ *y*, then **887** ◁ *p*. Hence we may assume *y* ∈ {0}, and thus *p* ∈ 8{0}7. But then, since the sum of the digits of *p* is 15, *p* is divisible by 3, so *p* cannot be prime.

Case 4: *p* ends with 9.

In this case we can write *p* = *x*9. If *x* contains 1, 2, 5, 7, or 8, then (respectively) **19** ◁ *p*, **29** ◁ *p*, **59** ◁ *p*, **79** ◁ *p*, or **89** ◁ *p*. Hence we may assume all digits of *x* are 0, 3, 4, 6, or 9.

If 44 ◁ *x*, then **449** ◁ *p*. Hence we may assume *x* contains zero or one 4's.

If x contains no 4's, then all digits of *x* are 0, 3, 6, or 9, and thus all digits of *p* are 0, 3, 6, or 9. But then, since the digits of *p* all have a common factor 3, *p* is divisible by 3, so *p* cannot be prime. Hence we may assume that *x* contains exactly one 4.

Case 4.1: *p* begins with 3.

In this case we can write *p* = 3*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. We must have **349** ◁ *p*.

Case 4.2: *p* begins with 4.

In this case we can write *p* = 4*y*9, where all digits of *y* are 0, 3, 6, or 9. If 0 ◁ *y*, then **409** ◁ *p*. If 3 ◁ *y*, then 43 ◁ *p*. If 9 ◁ *y*, then **499** ◁ *p*. Hence we may assume *y* ∈ {6}, and thus *p* ∈ 4{6}9. But then *p* is divisible by 7, since for *n* ≥ 0 we have 7 × 6<sub>*n*</sub>7 = 46<sub>*n*</sub>9.

Case 4.3: *p* begins with 6.

In this case we can write p = 6*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. If 0 ◁ *z*, then 409 ◁ *p*. If 3 ◁ *z*, then 43 ◁ *p*. If 6 ◁ *z*, then **6469** ◁ *p*. If 9 ◁ *z*, then 499 ◁ *p*. Hence we may assume *z* is empty.

If 3 ◁ *y*, then 349 ◁ *p*. If 9 ◁ *y*, then **6949** ◁ *p*. Hence we may assume all digits of *y* are 0 or 6.

If 06 ◁ *y*, then **60649** ◁ *p*. Hence we may assume *y* ∈ {6}{0}.

If 666 ◁ *y*, then **666649** ◁ *p*. If 00000 ◁ *y*, then **60000049** ◁ *p*. Hence we may assume *y* ∈ {*𝜆*, 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus *p* ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only **66000049** and **66600049** are primes.

Case 4.4: *p* begins with 9.

In this case we can write p = 9*y*4*z*9, where all digits of *y*, *z* are 0, 3, 6, or 9. If 0 ◁ *y*, then **9049** ◁ *p*. If 3 ◁ *y*, then 349 ◁ *p*. If 6 ◁ *y*, then **9649** ◁ *p*. If 9 ◁ *y*, then **9949** ◁ *p*. Hence we may assume *y* is empty.

If 0 ◁ *z*, then 409 ◁ *p*. If 3 ◁ *z*, then 43 ◁ *p*. If 9 ◁ *z*, then 499 ◁ *p*. Hence we may assume *z* ∈ {6}, and thus *p* ∈ 94{6}9, and the smallest prime *p* ∈ 94{6}9 is 946669.

I left it as an exercise for the reader to write the proof for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 12, of course, the proof for base *b* = 2 is trivial, since all primes *p* > 2 must start and end with 1 in base 2, thus we must have 11 ◁ *p*, however, for some bases *b* like 24 (the currently largest "proven" base *b*, including the primality proving for the primes in the set), it is almost impossible to write the proof by hand, since base *b* = 24 has too many (3409) minimal primes to write the proof, thus the C++ program code (for computer to compute (https://en.wikipedia.org/wiki/Computing) the proof) is made.

(in fact, the fully proof should also include the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set (like https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html (for the generalized repunit prime in base *b* = 3 with length 4177) and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html (for the generalized repunit prime in base *b* = 6 with length 6883) and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html (for the generalized repunit prime in base *b* = 7 with length 1699) and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html (for the generalized repunit prime in base *b* = 10 with length 1031) and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html (for the generalized repunit prime in base *b* = 11 with length 4801) and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html (for the generalized repunit prime in base *b* = 13 with length 991) and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html (for the generalized repunit prime in base *b* = 13 with length 1021) and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html (for the generalized repunit prime in base *b* = 13 with length 1193) and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html (for the generalized repunit prime in base *b* = 15 with length 2579) and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html (for the generalized repunit prime in base *b* = 22 with length 857) and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html (for the generalized repunit prime in base *b* = 31 with length 5581) and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html (for the generalized repunit prime in base *b* = 35 with length 1297) and https://stdkmd.net/nrr/pock/ (for the near-repdigit primes, although the primes 2×10<sup>1755</sup>−1 and 2×10<sup>3020</sup>−1 can be quickly proven prime using the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)) and http://xenon.stanford.edu/~tjw/pp/index.html (for the generalized repunit primes) and https://t5k.org/lists/single_primes/50005cert.txt (for the prime https://t5k.org/primes/page.php?id=12806) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_27669.out (for the large prime factor of 10<sup>27669</sup>+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_15093.out (for the prime 10<sup>15093</sup>+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_10393.out (for the large prime factor of 10<sup>10393</sup>+7) and https://homes.cerias.purdue.edu/~ssw/cun/third/proofs (for the larger prime factors of *b*<sup>*n*</sup>±1 with 2 ≤ *b* ≤ 12) and https://web.archive.org/web/20150911225651/https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0 (for the prime https://t5k.org/primes/page.php?id=11084) and https://web.archive.org/web/20170515153924/http://bitc.bme.emory.edu/~lzhou/blogs/?p=263 (for the primes corresponding to https://oeis.org/A181980) and https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html (for the prime https://t5k.org/primes/page.php?id=82858), or using an elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementation such as *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php)) and the compositeness proving for all proper subsequence of all primes in the set (usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) (usually to 10<sup>9</sup>, this will covered by sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) for the numbers > 10<sup>1000</sup>) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535) (usually base 2 and base 3)), but in the proof above we assume that we know whether a number is prime or not)

Problems about the digits of prime numbers have a long history, and many of them are still unsolved (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/). For example, are there infinitely many primes, all of whose base-10 digits are 1? Currently, there are only six such "repunits" (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) known, corresponding to (10<sup>*n*</sup>−1)/9 for *n* ∈ {2, 19, 23, 317, 1031, 49081, 86453} (references for recently proven prime with *n* = 49081 and *n* = 86453: https://mersenneforum.org/showpost.php?p=602219&postcount=35, https://mersenneforum.org/showpost.php?p=630711&postcount=236, https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/page.php?id=133761, https://t5k.org/primes/page.php?id=136044, https://kurtbeschorner.de/db-status-3-1M.htm, http://www.elektrosoft.it/matematica/repunit/repunit.htm, https://stdkmd.net/nrr/cert/Phi/Phi_49081_10.zip, https://stdkmd.net/nrr/cert/Phi/Phi_86453_10.zip, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.001, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.002, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.003, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.004, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.005, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.006). It seems likely that four more are given by *n* ∈ {109297, 270343, 5794777, 8177207}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php). This problem also exists for other bases, e.g. for base 12, there are only nine proven such numbers, corresponding to (12<sup>*n*</sup>−1)/11 for *n* ∈ {2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951}. It seems likely that three more are given by *n* ∈ {37573, 46889, 769543}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php).

Any repunit in any base *b* having a composite number of digits is necessarily composite. Only repunits (in any base *b*) having a prime number of digits might be prime. This is a necessary but not sufficient condition, e.g. 11111111111111111111111111111111111 (the repunit with 35 (= 5 × 7, which is composite) digits) = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001, since 35 = 5 × 7 = 7 × 5, and this repunit factorization does not depend on the base *b* in which the repunit is expressed. (note that the value of the repunit (in any base *b*) having 1 digit is 1, and 1 is not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one)).

A repunit (in any base *b*) with length *n* can be prime only if *n* is prime, since otherwise *b*<sup>*k*×*m*</sup>−1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/). In fact, if *n* = 2×*m* is even, then *b*<sup>2×*m*</sup>−1 = (*b*<sup>*m*</sup>−1) × (*b*<sup>*m*</sup>+1).

This is the list of the known generalized repunit (probable) primes in bases 2 ≤ *b* ≤ 36 (*italic* for unproven probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1)): (references: http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf))

|*b*|lengths of the generalized repunit primes base *b* (written in base 10)|search limit|*OEIS* sequence|
|---|---|---|---|
|2|2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ... (the Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://www.mersenne.org/, https://www.mersenne.org/primes/, https://www.mersenne.ca/, https://t5k.org/mersenne/), all are definitely primes, i.e. not merely probable primes)|64355023|https://oeis.org/A000043|
|3|3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, *43063*, *49681*, *57917*, *483611*, *877843*, *2215303*, *2704981*, *3598867*, ...|5000000|https://oeis.org/A028491|
|4|2 (this is all, since (4<sup>*n*</sup>−1)/3 = (2<sup>*n*</sup>−1) × (2<sup>*n*</sup>+1) / 3, and both 2<sup>*n*</sup>−1 and 2<sup>*n*</sup>+1 are > 3 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(infinity)|–|
|5|3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, *201359*, *396413*, *1888279*, *3300593*, ...|3300593|https://oeis.org/A004061|
|6|2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, *79987*, *608099*, *1365019*, ...|1365019|https://oeis.org/A004062|
|7|5, 13, 131, 149, 1699, 14221, *35201*, *126037*, *371669*, *1264699*, ...|1264699|https://oeis.org/A004063|
|8|3 (this is all, since (8<sup>*n*</sup>−1)/7 = (2<sup>*n*</sup>−1) × (4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 7, and both 2<sup>*n*</sup>−1 and 4<sup>*n*</sup>+2<sup>*n*</sup>+1 are > 7 for *n* > 3, thus this factorization is nontrivial for *n* > 3, it only remains to check the cases *n* = 2 and *n* = 3, but (8<sup>2</sup>−1)/7 = 9 = 3<sup>2</sup> is not prime)|(infinity)|–|
|9|not exist (since (9<sup>*n*</sup>−1)/8 = (3<sup>*n*</sup>−1) × (3<sup>*n*</sup>+1) / 8, and both 3<sup>*n*</sup>−1 and 3<sup>*n*</sup>+1 are > 8 for *n* > 2, thus this factorization is nontrivial for *n* > 2, it only remains to check the case *n* = 2, but (9<sup>2</sup>−1)/8 = 10 = 2 × 5 is not prime)|(infinity)|–|
|10|2, 19, 23, 317, 1031, 49081, 86453, *109297*, *270343*, *5794777*, *8177207*, ...|10800000|https://oeis.org/A004023|
|11|17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, *293831*, *1868983*, ...|1868983|https://oeis.org/A005808|
|12|2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, *37573*, *46889*, *769543*, ...|1000000|https://oeis.org/A004064|
|13|5, 7, 137, 283, 883, 991, 1021, 1193, 3671, *18743*, *31751*, *101089*, *1503503*, ...|1503503|https://oeis.org/A016054|
|14|3, 7, 19, 31, 41, 2687, *19697*, *59693*, *67421*, *441697*, ...|1000000|https://oeis.org/A006032|
|15|3, 43, 73, 487, 2579, 8741, *37441*, *89009*, *505117*, *639833*, ...|1000000|https://oeis.org/A006033|
|16|2 (this is all, since (16<sup>*n*</sup>−1)/15 = (4<sup>*n*</sup>−1) × (4<sup>*n*</sup>+1) / 15, and both 4<sup>*n*</sup>−1 and 4<sup>*n*</sup>+1 are > 15 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(infinity)|–|
|17|3, 5, 7, 11, 47, 71, 419, 4799, *35149*, *54919*, *74509*, *1990523*, ...|1990523|https://oeis.org/A006034|
|18|2, 25667, *28807*, *142031*, *157051*, *180181*, *414269*, *1270141*, ...|1270141|https://oeis.org/A133857|
|19|19, 31, 47, 59, 61, 107, 337, 1061, *9511*, *22051*, *209359*, ...|1000000|https://oeis.org/A006035|
|20|3, 11, 17, 1487, *31013*, *48859*, *61403*, *472709*, *984349*, ...|1000000|https://oeis.org/A127995|
|21|3, 11, 17, 43, 271, *156217*, *328129*, ...|1000000|https://oeis.org/A127996|
|22|2, 5, 79, 101, 359, 857, 4463, *9029*, *27823*, ...|1000000|https://oeis.org/A127997|
|23|5, 3181, *61441*, *91943*, *121949*, *221411*, ...|1000000|https://oeis.org/A204940|
|24|3, 5, 19, 53, 71, 653, 661, *10343*, *49307*, *115597*, *152783*, ...|1000000|https://oeis.org/A127998|
|25|not exist (since (25<sup>*n*</sup>−1)/24 = (5<sup>*n*</sup>−1) × (5<sup>*n*</sup>+1) / 24, and both 5<sup>*n*</sup>−1 and 5<sup>*n*</sup>+1 are > 24 for *n* > 2, thus this factorization is nontrivial for *n* > 2, it only remains to check the case *n* = 2, but (25<sup>2</sup>−1)/24 = 26 = 2 × 13 is not prime)|(infinity)|–|
|26|7, 43, 347, *12421*, *12473*, *26717*, ...|1000000|https://oeis.org/A127999|
|27|3 (this is all, since (27<sup>*n*</sup>−1)/26 = (3<sup>*n*</sup>−1) × (9<sup>*n*</sup>+3<sup>*n*</sup>+1) / 26, and both 3<sup>*n*</sup>−1 and 9<sup>*n*</sup>+3<sup>*n*</sup>+1 are > 26 for *n* > 3, thus this factorization is nontrivial for *n* > 3, it only remains to check the cases *n* = 2 and *n* = 3, but (27<sup>2</sup>−1)/26 = 28 = 2<sup>2</sup>×7 is not prime)|(infinity)|–|
|28|2, 5, 17, 457, 1423, *115877*, ...|1000000|https://oeis.org/A128000|
|29|5, 151, 3719, *49211*, *77237*, ...|1000000|https://oeis.org/A181979|
|30|2, 5, 11, 163, 569, 1789, *8447*, *72871*, *78857*, *82883*, ...|1000000|https://oeis.org/A098438|
|31|7, 17, 31, 5581, *9973*, *54493*, *101111*, *535571*, ...|1000000|https://oeis.org/A128002|
|32|not exist (since (32<sup>*n*</sup>−1)/31 = (2<sup>*n*</sup>−1) × (16<sup>*n*</sup>+8<sup>*n*</sup>+4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 31, and both 2<sup>*n*</sup>−1 and 16<sup>*n*</sup>+8<sup>*n*</sup>+4<sup>*n*</sup>+2<sup>*n*</sup>+1 are > 31 for *n* > 5, thus this factorization is nontrivial for *n* > 5, it only remains to check the cases *n* = 2 and *n* = 3 and *n* = 4 and *n* = 5, but (32<sup>2</sup>−1)/31 = 33 = 3 × 11 and (32<sup>3</sup>−1)/31 = 1057 = 7 × 151 and (32<sup>4</sup>−1)/31 = 33825 = 3 × 5<sup>2</sup> × 11 × 41 and (32<sup>5</sup>−1)/31 = 1082401 = 601 × 1801 are not primes)|(infinity)|–|
|33|3, 197, 3581, *6871*, *183661*, ...|1000000|https://oeis.org/A209120|
|34|13, 1493, *5851*, *6379*, *125101*, ...|1000000|https://oeis.org/A185073|
|35|313, 1297, *568453*, ...|1000000|https://oeis.org/A348170|
|36|2 (this is all, since (36<sup>*n*</sup>−1)/35 = (6<sup>*n*</sup>−1) × (6<sup>*n*</sup>+1) / 35, and both 6<sup>*n*</sup>−1 and 6<sup>*n*</sup>+1 are > 35 for *n* > 2, thus this factorization is nontrivial for *n* > 2)|(infinity)|–|

Determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering is in general unsolvable, and can be difficult even when this set is relatively simple (such as the base *b* representations of the prime numbers > *b*), also, determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering may be an open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/) or NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) or an undecidable problem (https://en.wikipedia.org/wiki/Undecidable_problem, https://mathworld.wolfram.com/Undecidable.html), or an example of Gödel's incompleteness theorems (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems, https://mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html) (like the continuum hypothesis (https://en.wikipedia.org/wiki/Continuum_hypothesis, https://mathworld.wolfram.com/ContinuumHypothesis.html) and the halting problem (https://en.wikipedia.org/wiki/Halting_problem, https://mathworld.wolfram.com/HaltingProblem.html)), or as hard as the unsolved problems in mathematics, such as the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) and the *abc* conjecture (https://en.wikipedia.org/wiki/Abc_conjecture, https://mathworld.wolfram.com/abcConjecture.html, http://www.numericana.com/answer/open.htm#abc) and the Schinzel's hypothesis *H* (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h), which are the three famous hard problems in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html).

The following is a "semi-algorithm" (https://en.wikipedia.org/wiki/Semi-algorithm) that is guaranteed to produce the minimal elements of a arbitrary set of strings under the subsequence ordering, but it is not so easy to implement:

1. *M* := *∅* 
2. while (*L* ≠ *∅*) do
3. choose *x*, a shortest string in *L*
4. *M* := *M* ∪ {*x*}
5. *L* := *L* − *sup*({*x*})

In practice, for arbitrary *L*, we cannot feasibly carry out step 5. Instead, we work with *L*', some regular overapproximation to *L*, until we can show *L*' = *∅* (which implies *L* = *∅*). In practice, *L*' is usually chosen to be a finite union of sets of the form *L*<sub>1</sub>{*L*<sub>2</sub>}*L*<sub>3</sub>, where each of *L*<sub>1</sub>, *L*<sub>2</sub>, *L*<sub>3</sub> is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime > *b* or not.

To solve this problem (i.e. to compute (https://en.wikipedia.org/wiki/Computing) the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering), we need to determine whether a given family contains a prime. In practice, if family *x*{*Y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *Y* is a set of digits in base *b*) could not be ruled out as only containing composites and *Y* contains two or more digits, then a relatively small prime > *b* could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2<sup>*n*−2</sup> strings of length *n* in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case *Y* contains only one digit, this family is of the form *x*{*y*}*z*, and there is only a single string of each length > (the length of *x* + the length of *z*), and it is not known if the following decision problem (https://en.wikipedia.org/wiki/Decision_problem, https://mathworld.wolfram.com/DecisionProblem.html) is recursively solvable:

Problem: Given strings *x*, *z* (may be empty), a digit *y*, and a base *b*, does there exist a prime number whose base-*b* expansion is of the form *xy*<sub>*n*</sub>*z* for some *n* ≥ 0? (If we say "yes", then we should find a such prime (the smallest such prime may be very large, e.g. > 10<sup>25000</sup>, and if so, then we should use (probable) primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program) such as *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) or *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to find it, and before using these programs, we should use sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) programs (https://www.rieselprime.de/ziki/Sieving_program) such as *srsieve* (or *sr*1/2/5*sieve*) (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to remove the numbers either having small prime factors or having algebraic factors) and prove its primality (by *N*−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) or *N*+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) or elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementation such as *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php)) (and if we want to solve the problem in this project, we should check whether this prime is the smallest such prime or not, i.e. prove all smaller numbers of the form *xy*<sub>*n*</sub>*z* with *n* ≥ 0 are composite, usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535)), and if we say "no", then we should prove that such prime does not exist, may by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://web.archive.org/web/20060925100410/http://www.glasgowg43.freeserve.co.uk/siernums.htm, https://web.archive.org/web/20061116164533/http://www.glasgowg43.freeserve.co.uk/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) *S* of primes *p* such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of *S* (this is equivalent to finding a positive integer *N* such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *N*, this *N* is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b*, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11<sup>*n*</sup>−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime *p* such that all numbers in a given family are not *p*-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>), or combine of them (https://mersenneforum.org/showthread.php?t=11143, https://mersenneforum.org/showthread.php?t=10279, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://oeis.org/A213353, https://oeis.org/A233469))

An algorithm to solve this problem, for example, would allow us to decide if there are any additional Fermat primes (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, http://www.fermatsearch.org/, https://64ordle.au/fermat/) (of the form 2<sup>2<sup>*n*</sup></sup>+1) other than the known ones (corresponding to *n* = 0, 1, 2, 3, 4). To see this, take *b* = 2, *x* = 1, *y* = 0, and *z* = 0<sub>16</sub>1. Since if 2<sup>*n*</sup>+1 is prime then *n* must be a power of two (http://yves.gallot.pagesperso-orange.fr/primes/math.html), a prime of the form *xy*<sub>*n*</sub>*z* in base *b* must be a new Fermat prime. Besides, it would allow us to decide if there are infinitely many Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://www.mersenne.org/, https://www.mersenne.org/primes/, https://www.mersenne.ca/, https://t5k.org/mersenne/) (of the form 2<sup>*p*</sup>−1 with prime *p*). To see this, take *b* = 2, *x* = *𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string)), y = 1, and z = 1<sub>*n*+1</sub>, where *n* is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2<sup>*n*</sup>−1 is prime then *n* must be a prime (https://t5k.org/notes/proofs/Theorem2.html), a prime of the form *xy*<sub>*n*</sub>*z* in base *b* must be a new Mersenne prime. Also, it would allow us to decide whether 78557 is the smallest Sierpinski number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076336) and whether 509203 is the smallest Riesel number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076337, https://oeis.org/A101036), etc.

(Currently, whether 65537 is the largest Fermat prime, whether there are infinitely many Mersenne primes, whether 78557 is the smallest Sierpinski number, whether 509203 is the smallest Riesel number, are all unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/))

Also, there are some examples in decimal (i.e. base *b* = 10): (references: https://stdkmd.net/nrr/prime/primecount.htm, https://stdkmd.net/nrr/prime/primecount.txt, https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt) (see https://sites.google.com/view/smallest-quasi-repdigit-primes for more examples)

|*x*|*y*|*z*|answer|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest allowed *n* in the table (usually *n* = 0, unless other *n* are mentioned))|
|---|---|---|---|---|
|5028|0|1|Yes! But the smallest such prime is very large, it is 50280<sub>83981</sub>1, its algebraic form is 5028×10<sup>83982</sup>+1, its *factordb* entry is http://factordb.com/index.php?id=1100000000765961536, it can be proven prime by the *N*−1 primality proving (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1)|http://factordb.com/index.php?query=5028*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7018|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 70189<sub>881309</sub>, its algebraic form is 7019×10<sup>881309</sup>−1, its *factordb* entry is http://factordb.com/index.php?id=1100000000628445542, it can be proven prime by the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)|http://factordb.com/index.php?query=7019*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9175|0|1|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 9175×10<sup>*n*+1</sup>+1, it is divisible by 7 if *n* == 3 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 2 mod 3|http://factordb.com/index.php?query=9175*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10175|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 10176×10<sup>*n*</sup>−1, it is divisible by 7 if *n* == 1 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 5 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=10176*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7666|0|1|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 7666×10<sup>*n*+1</sup>+1, and if such prime exists, then it must have *n* > 2000000|http://factordb.com/index.php?query=7666*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|4420|9|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 4421×10<sup>*n*</sup>−1, and if such prime exists, then it must have *n* > 2000000|http://factordb.com/index.php?query=4421*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1|7|Yes! But the smallest such prime is very large, it is 71<sub>10905</sub>7, its algebraic form is (64×10<sup>10906</sup>+53)/9, its *factordb* entry is http://factordb.com/index.php?id=1000000000008860930, its primality certificate is http://factordb.com/cert.php?id=1000000000008860930 and https://stdkmd.net/nrr/cert/7/71117_10906.zip|http://factordb.com/index.php?query=%2864*10%5E%28n%2B1%29%2B53%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|4|9|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (85×10<sup>*n*+1</sup>+41)/9, it is divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6|http://factordb.com/index.php?query=%2885*10%5E%28n%2B1%29%2B41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|1|0 (at least two 0)|1|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 10<sup>*n*+1</sup>+1, and if such prime exists, then it must have *n* ≥ 2147483648|http://factordb.com/index.php?query=10%5E%28n%2B1%29%2B1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|56|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 561<sub>18470</sub>, its algebraic form is (505×10<sup>18470</sup>−1)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301454592, its primality certificate is http://factordb.com/cert.php?id=1100000000301454592 and https://stdkmd.net/nrr/cert/5/56111_18470.zip|http://factordb.com/index.php?query=%28505*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|38|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is either divisible by some element of {3,37} or has a difference-of-two-cubes factorization, since the algebraic form of this family is (343×10<sup>*n*</sup>−1)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 37 if *n* == 2 mod 3, and can be factored to (343×10<sup>*n*</sup>−1)/9 = (7×10<sup>*n*/3</sup>−1) × (49×10<sup>2×*n*/3</sup>+7×10<sup>*n*/3</sup>+1) / 9 if *n* == 0 mod 3|http://factordb.com/index.php?query=%28343*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|176|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (1585×10<sup>*n*</sup>−1)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 3 mod 6|http://factordb.com/index.php?query=%281585*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|603|1|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (5428×10<sup>*n*</sup>−1)/9, and if such prime exists, then it must have *n* > 300000|http://factordb.com/index.php?query=%285428*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|2|99|Yes! But the smallest such prime is very large, it is 2<sub>19151</sub>99, its algebraic form is (2×10<sup>19153</sup>+691)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301493137, its primality certificate is http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.zip|http://factordb.com/index.php?query=%282*10%5E%28n%2B2%29%2B691%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|5 (at least one 5)|37|No! All numbers of this form is divisible by some element of {3,7,13,37}, since the algebraic form of this family is (5×10<sup>*n*+2</sup>−167)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 2 mod 6, divisible by 13 if *n* == 5 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=%285*10%5E%28n%2B2%29-167%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|44|9 (at least two 9)|*𝜆* (the empty string (https://en.wikipedia.org/wiki/Empty_string))|Yes! But the smallest such prime is very large, it is 449<sub>11959</sub>, its algebraic form is 45×10<sup>11959</sup>−1, its *factordb* entry is http://factordb.com/index.php?id=1100000000291927010, it can be proven prime by the *N*+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)|http://factordb.com/index.php?query=45*10%5En-1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|421|0|3|Almost surely Yes! But the smallest such (probable) prime is very large, it is 4210<sub>16019</sub>3, its algebraic form is 421×10<sup>16020</sup>+3, its *factordb* entry is http://factordb.com/index.php?id=1100000002392921307, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=421*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|404|0|3|No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 404×10<sup>*n*+1</sup>+3, it is divisible by 7 if *n* == 5 mod 6, divisible by 11 if *n* == 0 mod 2, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 0 mod 3|http://factordb.com/index.php?query=404*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|99|4|99|Almost surely Yes! But the smallest such (probable) prime is very large, it is 994<sub>34019</sub>99, its algebraic form is (895×10<sup>34021</sup>+491)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000002454717990, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=%28895*10%5E%28n%2B2%29%2B491%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|2|5|67|Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (23×10<sup>*n*+2</sup>+103)/9, and if such prime exists, then it must have *n* > 30000|http://factordb.com/index.php?query=%2823*10%5E%28n%2B2%29%2B103%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1|83|Almost surely Yes! But the smallest such (probable) prime is very large, it is 71<sub>18466</sub>83, its algebraic form is (64×10<sup>18468</sup>+647)/9, its *factordb* entry is http://factordb.com/index.php?id=1100000000301454024, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known *definitely* prime)|http://factordb.com/index.php?query=%2864*10%5E%28n%2B2%29%2B647%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|3|2|01|No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (29×10<sup>*n*+2</sup>−191)/9, it is divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 0 mod 6, divisible by 11 if *n* == 1 mod 2, divisible by 13 if *n* == 2 mod 6|http://factordb.com/index.php?query=%2829*10%5E%28n%2B2%29-191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

**My conjecture: If family *xy*<sub>***n***</sub>*z* (with fixed strings *x*, *z* (may be empty), fixed digit *y*, and variable *n*) in base *b* (with fixed *b* ≥ 2) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family *xy*<sub>***n***</sub>*z* in base *b* contains infinitely many primes (this is equivalent to: If form (*a*×*b*<sup>***n***</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) (with fixed integers *a* ≥ 1, *b* ≥ 2, *c* ≠ 0 (with *gcd*(*a*,*c*) = 1 and *gcd*(*b*,*c*) = 1), and variable *n*) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (*a*×*b*<sup>***n***</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) contains infinitely many primes)**

(this conjecture (https://en.wikipedia.org/wiki/Conjecture, https://t5k.org/glossary/xpage/Conjecture.html, https://mathworld.wolfram.com/Conjecture.html) is very important for the problem in this project, since if this conjecture is in fact false, then there will may be some unsolved families which in fact contain no primes, thus the problem in this project in corresponding bases *b* will may be unsolvable (at least in current technology, unless someone find a new theorem (i.e. other than covering congruence, algebraic factorization, or combine of them) to prove that some families contain no primes, but I do not think that this is possible), however, this conjecture is currently to far to prove, much far than the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and even the Schinzel's hypothesis *H* (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h), besides, this conjecture is reasonable, since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for *s* = 509203 and *s* = −78557, *s* = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and *s* = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://mersenneforum.org/showpost.php?p=564786&postcount=3, https://mersenneforum.org/showpost.php?p=461665&postcount=7, https://mersenneforum.org/showpost.php?p=625978&postcount=1027, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base *b* = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base *b* = 10) and https://mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base *b* = 3)), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) *n*-digit base *b* number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/*n* (more accurately, the chance is approximately 1/(*n*×*ln*(*b*)), where *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base *e* = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html)). If one conjectures the numbers *x*{*y*}*z* behave similarly (i.e. the numbers *x*{*y*}*z* is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form *x*{*y*}*z* (of course, this does not always happen, since some *x*{*y*}*z* families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=421186&postcount=19, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://mersenneforum.org/showpost.php?p=564786&postcount=3, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), but it is at least a reasonable conjecture in the absence of evidence to the contrary), there are approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) (*e*<sup>*γ*</sup>×*W*×*N*−1/1−1/2−1/3−...−1/(*length*(*x*)+*length*(*z*)−1))/*ln*(*b*) primes in the family *x*{*y*}*z* in base *b* with length ≤ *N* (where *e* = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), *γ* = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), *W* is the Nash weight (or difficulty) of the family *x*{*y*}*z* in base *b* (*W* = 0 if and only if the family *x*{*y*}*z* in base *b* can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html) (i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html))).

(this conjecture is for exponential sequences (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) (with fixed integers *a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*, *c*) = 1, *gcd*(*b*, *c*) = 1, and variable *n*), there is also a similar conjecture for polynomial sequences (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) *a*<sub>0</sub>+*a*<sub>1</sub>*x*+*a*<sub>2</sub>*x*<sup>2</sup>+*a*<sub>3</sub>*x*<sup>3</sup>+...+*a*<sub>*n*−1</sub>*x*<sup>*n*−1</sup>+*a*<sub>*n*</sub>*x*<sup>*n*</sup> (with fixed *n*, *a*<sub>0</sub>, *a*<sub>1</sub>, *a*<sub>2</sub>, ..., *a*<sub>*n*</sub> and variable *x*): the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html), the condition is similar to this conjecture (divisible by small primes and algebraic factorizations), the main difference is that polynomial sequence cannot have a covering congruence with > 1 primes, nor have a combine of covering congruence and algebraic factorization)

This conjecture will imply:

* There are infinitely many Mersenne primes (i.e. primes of the form 2<sup>*p*</sup>−1 with prime *p*) (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://www.mersenne.org/, https://www.mersenne.org/primes/, https://www.mersenne.ca/, https://t5k.org/mersenne/) (https://oeis.org/A001348, https://oeis.org/A000668, https://oeis.org/A000043)
* There are infinitely many Fermat primes (i.e. primes of the form 2<sup>2<sup>*n*</sup></sup>+1) (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, http://www.fermatsearch.org/, https://64ordle.au/fermat/) (https://oeis.org/A000215, https://oeis.org/A019434)
* There are infinitely many generalized repunit primes (i.e. primes of the form (*b*<sup>*p*</sup>−1)/(*b*−1) with prime *p*) (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) to every base *b* ≥ 2 which is not a perfect power (i.e. of the form *m*<sup>*r*</sup> with *r* > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) (https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A246005, https://oeis.org/A065854, https://oeis.org/A279068, https://oeis.org/A065813, https://oeis.org/A128164, https://oeis.org/A285642)
* There are infinitely many generalized Wagstaff primes (i.e. primes of the form (*b*<sup>*p*</sup>+1)/(*b*+1) with odd prime *p*) (https://en.wikipedia.org/wiki/Wagstaff_prime, https://t5k.org/glossary/xpage/WagstaffPrime.html, https://mathworld.wolfram.com/WagstaffPrime.html, https://pzktupel.de/Primetables/TableWagstaff.php, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://pzktupel.de/Primetables/TableWagstaffGen.txt, https://web.archive.org/web/20211031110623/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/, http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/table/488, https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://t5k.org/top20/page.php?id=67, https://oeis.org/A007583, https://oeis.org/A000979, https://oeis.org/A000978, https://oeis.org/A059054, https://oeis.org/A059055) to every base *b* ≥ 2 which is neither a perfect odd power (i.e. of the form *m*<sup>*r*</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*<sup>4</sup> (https://oeis.org/A141046) (https://oeis.org/A084742, https://oeis.org/A084741, https://oeis.org/A126659, https://oeis.org/A065507)
* There are infinitely many generalized Fermat primes (i.e. primes of the form *b*<sup>2<sup>*n*</sup></sup>+1 with even *b*) (https://t5k.org/glossary/xpage/GeneralizedFermatNumber.html, https://t5k.org/glossary/xpage/GeneralizedFermatPrime.html, https://www.rieselprime.de/ziki/Generalized_Fermat_number, https://mathworld.wolfram.com/GeneralizedFermatNumber.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF09.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF10.php, https://pzktupel.de/Primetables/TableFermatGF11.php, https://pzktupel.de/Primetables/TableFermatGF12.php, https://pzktupel.de/Primetables/TableFermatGF13.php, https://pzktupel.de/Primetables/TableFermatGF14.php, https://pzktupel.de/Primetables/TableFermatGF15.php, https://pzktupel.de/Primetables/TableFermatGF16.php, https://pzktupel.de/Primetables/TableFermatGF17.php, https://pzktupel.de/Primetables/TableFermatGF1820.php, http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, http://yves.gallot.pagesperso-orange.fr/primes/index.html, http://yves.gallot.pagesperso-orange.fr/primes/results.html, http://yves.gallot.pagesperso-orange.fr/primes/stat.html, http://www.primegrid.com/forum_thread.php?id=3980, http://www.primegrid.com/stats_genefer.php, https://t5k.org/top20/page.php?id=12, http://www.prothsearch.com/fermat.html, http://www.prothsearch.com/GFN06.html, http://www.prothsearch.com/GFN10.html, http://www.prothsearch.com/GFN12.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html, https://oeis.org/A006314, https://oeis.org/A006313, https://oeis.org/A006315, https://oeis.org/A006316, https://oeis.org/A056994, https://oeis.org/A056995, https://oeis.org/A057465, https://oeis.org/A057002, https://oeis.org/A088361, https://oeis.org/A088362, https://oeis.org/A226528, https://oeis.org/A226529, https://oeis.org/A226530, https://oeis.org/A251597, https://oeis.org/A253854, https://oeis.org/A244150, https://oeis.org/A243959, https://oeis.org/A321323) to every even base *b* ≥ 2 which is not a perfect odd power (i.e. of the form *m*<sup>*r*</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) (https://oeis.org/A228101, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A123669)
* There are infinitely many generalized half-Fermat primes (i.e. primes of the form (*b*<sup>2<sup>*n*</sup></sup>+1)/2 with odd *b*) (http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German), http://www.prothsearch.com/GFN03.html, http://www.prothsearch.com/GFN05.html, http://www.prothsearch.com/GFN07.html, http://www.prothsearch.com/GFN11.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html) to every odd base *b* ≥ 2 which is not a perfect odd power (i.e. of the form *m*<sup>*r*</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265)
* There are infinitely many Williams primes of the first kind (i.e. primes of the form (*b*−1)×*b*<sup>*n*</sup>−1) (https://harvey563.tripod.com/wills.txt, https://www.rieselprime.de/ziki/Williams_prime_MM_least, https://www.rieselprime.de/ziki/Williams_prime_MM_table, https://pzktupel.de/Primetables/TableWilliams1.php, https://sites.google.com/view/williams-primes, http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf), https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf), http://www.bitman.name/math/table/484 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A122396)
* There are infinitely many Williams primes of the second kind (i.e. primes of the form (*b*−1)×*b*<sup>*n*</sup>+1) (https://www.rieselprime.de/ziki/Williams_prime_MP_least, https://www.rieselprime.de/ziki/Williams_prime_MP_table, https://pzktupel.de/Primetables/TableWilliams2.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/477 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A305531, https://oeis.org/A087139) **(warning: this may be false, (*b*−1)×*b*<sup>***n***</sup>+1 may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when *b*−1 is either a perfect odd power (i.e. of the form *m*<sup>***r***</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×*m*<sup>**4**</sup> (https://oeis.org/A141046), but the smallest such base *b* will be very large, however, this is at least true for bases *b* such that *b*−1 is neither a perfect odd power (i.e. of the form *m*<sup>***r***</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*<sup>**4**</sup> (https://oeis.org/A141046), also at least true for bases *b* ≤ 10<sup>**6**</sup>)**
* There are infinitely many Williams primes of the third kind (i.e. primes of the form (*b*+1)×*b*<sup>*n*</sup>−1) (https://www.rieselprime.de/ziki/Williams_prime_PM_least, https://www.rieselprime.de/ziki/Williams_prime_PM_table, https://pzktupel.de/Primetables/TableWilliams3.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/471 (in Italian)) to every base *b* ≥ 2
* There are infinitely many Williams primes of the fourth kind (i.e. primes of the form (*b*+1)×*b*<sup>*n*</sup>+1) (https://www.rieselprime.de/ziki/Williams_prime_PP_least, https://www.rieselprime.de/ziki/Williams_prime_PP_table, https://pzktupel.de/Primetables/TableWilliams4.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/474 (in Italian)) to every base *b* ≥ 2 which is not == 1 mod 3 **(warning: this may be false, (*b*+1)×*b*<sup>***n***</sup>+1 may be able to be proven to only contain composites by covering congruence, like the case of 2×*b*<sup>***n***</sup>+1 and *b*<sup>***n***</sup>+2 for *b* = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases *b* ≤ 10<sup>**6**</sup>)**
* There are infinitely many dual Williams primes of the first kind (i.e. primes of the form *b*<sup>*n*</sup>−(*b*−1)) (https://sites.google.com/view/williams-primes, https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435 (in Italian)) to every base *b* ≥ 2 (https://oeis.org/A113516, https://oeis.org/A343589)
* There are infinitely many dual Williams primes of the second kind (i.e. primes of the form *b*<sup>*n*</sup>+(*b*−1)) (https://sites.google.com/view/williams-primes) to every base *b* ≥ 2 (https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179) **(warning: this may be false, *b*<sup>***n***</sup>+(*b*−1) may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when *b*−1 is either a perfect odd power (i.e. of the form *m*<sup>***r***</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×*m*<sup>**4**</sup> (https://oeis.org/A141046), but the smallest such base *b* will be very large, however, this is at least true for bases *b* such that *b*−1 is neither a perfect odd power (i.e. of the form *m*<sup>***r***</sup> with odd *r* > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×*m*<sup>**4**</sup> (https://oeis.org/A141046), also at least true for bases *b* ≤ 10<sup>**6**</sup>)**
* There are infinitely many dual Williams primes of the third kind (i.e. primes of the form *b*<sup>*n*</sup>−(*b*+1)) (https://sites.google.com/view/williams-primes) to every base *b* ≥ 2 (https://oeis.org/A178250)
* There are infinitely many dual Williams primes of the fourth kind (i.e. primes of the form *b*<sup>*n*</sup>+(*b*+1)) (https://sites.google.com/view/williams-primes) to every base *b* ≥ 2 which is not == 1 mod 3 (https://oeis.org/A346149, https://oeis.org/A346154) **(warning: this may be false, *b*<sup>***n***</sup>+(*b*+1) may be able to be proven to only contain composites by covering congruence, like the case of 2×*b*<sup>***n***</sup>+1 and *b*<sup>***n***</sup>+2 for *b* = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases *b* ≤ 10<sup>**6**</sup>)**
* 78557 is the smallest Sierpinski number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247)
* 509203 is the smallest Riesel number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2.zip, http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050)
* 271129 is the second-smallest Sierpinski number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247)
* 762701 is the second-smallest Riesel number (i.e. odd numbers *k* such that *k*×2<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2.zip, http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050)
* 125050976086 is the smallest generalized Sierpinski number to base 3 (i.e. numbers *k* such that *gcd*(*k*+1, 3−1) = 1 and *k*×3<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/remain-sierp-base3.zip, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-sierp-base3-gt-25K.zip, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A291437, https://oeis.org/A291438)
* 63064644938 is the smallest generalized Riesel number to base 3 (i.e. numbers *k* such that *gcd*(*k*−1, 3−1) = 1 and *k*×3<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/remain-riesel-base3.zip, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base3-gt-25K.zip, https://oeis.org/A273987)
* 66741 is the smallest generalized Sierpinski number to base 4 (i.e. numbers *k* such that *gcd*(*k*+1, 4−1) = 1 and *k*×4<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base4.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A251057, https://oeis.org/A256002, http://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.primepuzzles.net/problems/prob_036.htm)
* 39939 is the smallest non-square generalized Riesel number to base 4 (i.e. numbers *k* such that *gcd*(*k*−1, 4−1) = 1 and *k*×4<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base4.txt, https://oeis.org/A273987, https://oeis.org/A251757, http://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.primepuzzles.net/problems/prob_036.htm)
* 159986 is the smallest generalized Sierpinski number to base 5 (i.e. numbers *k* such that *gcd*(*k*+1, 5−1) = 1 and *k*×5<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base5.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html, https://oeis.org/A345698)
* 346802 is the smallest generalized Riesel number to base 5 (i.e. numbers *k* such that *gcd*(*k*−1, 5−1) = 1 and *k*×5<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base5.zip, https://oeis.org/A273987, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html, https://oeis.org/A345403)
* 174308 is the smallest generalized Sierpinski number to base 6 (i.e. numbers *k* such that *gcd*(*k*+1, 6−1) = 1 and *k*×6<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base6.zip, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A244549, https://oeis.org/A250204)
* 84687 is the smallest generalized Riesel number to base 6 (i.e. numbers *k* such that *gcd*(*k*−1, 6−1) = 1 and *k*×6<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base6.txt, https://oeis.org/A273987, https://oeis.org/A244351, https://oeis.org/A250205)
* 1112646039348 is the smallest generalized Sierpinski number to base 7 (i.e. numbers *k* such that *gcd*(*k*+1, 7−1) = 1 and *k*×7<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base7-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-prime.htm, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 408034255082 is the smallest generalized Riesel number to base 7 (i.e. numbers *k* such that *gcd*(*k*−1, 7−1) = 1 and *k*×7<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base7-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base7-gt-25K.txt, https://oeis.org/A273987)
* 47 is the smallest non-cube generalized Sierpinski number to base 8 (i.e. numbers *k* such that *gcd*(*k*+1, 8−1) = 1 and *k*×8<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base8.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 31×8<sup>20</sup>+1)**
* 14 is the smallest non-cube generalized Riesel number to base 8 (i.e. numbers *k* such that *gcd*(*k*−1, 8−1) = 1 and *k*×8<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base8.txt, https://oeis.org/A273987) **(solved, largest prime is 11×8<sup>18</sup>−1)**
* 2344 is the smallest generalized Sierpinski number to base 9 (i.e. numbers *k* such that *gcd*(*k*+1, 9−1) = 1 and *k*×9<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base9.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 74 is the smallest non-square generalized Riesel number to base 9 (i.e. numbers *k* such that *gcd*(*k*−1, 9−1) = 1 and *k*×9<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base9.txt, https://oeis.org/A273987) **(solved, largest prime is 24×9<sup>8</sup>−1)**
* 9175 is the smallest generalized Sierpinski number to base 10 (i.e. numbers *k* such that *gcd*(*k*+1, 10−1) = 1 and *k*×10<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base10.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A243969)
* 10176 is the smallest generalized Riesel number to base 10 (i.e. numbers *k* such that *gcd*(*k*−1, 10−1) = 1 and *k*×10<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base10.txt, https://oeis.org/A273987, https://oeis.org/A243974)
* 1490 is the smallest generalized Sierpinski number to base 11 (i.e. numbers *k* such that *gcd*(*k*+1, 11−1) = 1 and *k*×11<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base11.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 958×11<sup>300544</sup>+1)**
* 862 is the smallest generalized Riesel number to base 11 (i.e. numbers *k* such that *gcd*(*k*−1, 11−1) = 1 and *k*×11<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base11.txt, https://oeis.org/A273987) **(solved, largest prime is 62×11<sup>26202</sup>−1)**
* 521 is the smallest generalized Sierpinski number to base 12 (i.e. numbers *k* such that *gcd*(*k*+1, 12−1) = 1 and *k*×12<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base12.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 376 is the smallest non-(*m*<sup>2</sup> with *m* == 5, 8 mod 13 or 3×*m*<sup>2</sup> with *m* == 3, 10 mod 13) generalized Riesel number to base 12 (i.e. numbers *k* such that *gcd*(*k*−1, 12−1) = 1 and *k*×12<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base12.txt, https://oeis.org/A273987) **(solved, largest prime is 157×12<sup>285</sup>−1)**
* 132 is the smallest generalized Sierpinski number to base 13 (i.e. numbers *k* such that *gcd*(*k*+1, 13−1) = 1 and *k*×13<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base13.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 48×13<sup>6267</sup>+1)**
* 302 is the smallest generalized Riesel number to base 13 (i.e. numbers *k* such that *gcd*(*k*−1, 13−1) = 1 and *k*×13<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base13.txt, https://oeis.org/A273987) **(solved, largest prime is 288×13<sup>109217</sup>−1)**
* 4 is the smallest generalized Sierpinski number to base 14 (i.e. numbers *k* such that *gcd*(*k*+1, 14−1) = 1 and *k*×14<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base14.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 1×14<sup>2</sup>+1)**
* 4 is the smallest generalized Riesel number to base 14 (i.e. numbers *k* such that *gcd*(*k*−1, 14−1) = 1 and *k*×14<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base14.txt, https://oeis.org/A273987) **(solved, largest prime is 2×14<sup>4</sup>−1)**
* 91218919470156 is the smallest generalized Sierpinski number to base 15 (i.e. numbers *k* such that *gcd*(*k*+1, 15−1) = 1 and *k*×15<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base15-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-prime.htm, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 36370321851498 is the smallest generalized Riesel number to base 15 (i.e. numbers *k* such that *gcd*(*k*−1, 15−1) = 1 and *k*×15<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base15-10M.zip, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-prime.htm, https://oeis.org/A273987)
* 66741 is the smallest non-(4×*m*<sup>4</sup>) generalized Sierpinski number to base 16 (i.e. numbers *k* such that *gcd*(*k*+1, 16−1) = 1 and *k*×16<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base16.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 33965 is the smallest non-square generalized Riesel number to base 16 (i.e. numbers *k* such that *gcd*(*k*−1, 16−1) = 1 and *k*×16<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base16.txt, https://oeis.org/A273987)
* 278 is the smallest generalized Sierpinski number to base 17 (i.e. numbers *k* such that *gcd*(*k*+1, 17−1) = 1 and *k*×17<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base17.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 86 is the smallest generalized Riesel number to base 17 (i.e. numbers *k* such that *gcd*(*k*−1, 17−1) = 1 and *k*×17<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base17.txt, https://oeis.org/A273987) **(solved, largest prime is 44×17<sup>6488</sup>−1)**
* 398 is the smallest generalized Sierpinski number to base 18 (i.e. numbers *k* such that *gcd*(*k*+1, 18−1) = 1 and *k*×18<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base18.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 246 is the smallest generalized Riesel number to base 18 (i.e. numbers *k* such that *gcd*(*k*−1, 18−1) = 1 and *k*×18<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base18.txt, https://oeis.org/A273987) **(solved, largest prime is 151×18<sup>418</sup>−1)**
* 765174 is the smallest generalized Sierpinski number to base 19 (i.e. numbers *k* such that *gcd*(*k*+1, 19−1) = 1 and *k*×19<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base19.zip, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 1119866 is the smallest non-(*m*<sup>2</sup> with *m* == 2, 3 mod 5 or 19×*m*<sup>2</sup> with *m* == 2, 3 mod 13) generalized Riesel number to base 19 (i.e. numbers *k* such that *gcd*(*k*−1, 19−1) = 1 and *k*×19<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base19.zip, https://oeis.org/A273987)
* 8 is the smallest generalized Sierpinski number to base 20 (i.e. numbers *k* such that *gcd*(*k*+1, 20−1) = 1 and *k*×20<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base20.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 6×20<sup>15</sup>+1)**
* 8 is the smallest generalized Riesel number to base 20 (i.e. numbers *k* such that *gcd*(*k*−1, 20−1) = 1 and *k*×20<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base20.txt, https://oeis.org/A273987) **(solved, largest prime is 2×20<sup>10</sup>−1)**
* 1002 is the smallest generalized Sierpinski number to base 21 (i.e. numbers *k* such that *gcd*(*k*+1, 21−1) = 1 and *k*×21<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base21.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 118×21<sup>19849</sup>+1)**
* 560 is the smallest generalized Riesel number to base 21 (i.e. numbers *k* such that *gcd*(*k*−1, 21−1) = 1 and *k*×21<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base21.txt, https://oeis.org/A273987) **(solved, largest prime is 64×21<sup>2867</sup>−1)**
* 6694 is the smallest generalized Sierpinski number to base 22 (i.e. numbers *k* such that *gcd*(*k*+1, 22−1) = 1 and *k*×22<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base22.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 4461 is the smallest generalized Riesel number to base 22 (i.e. numbers *k* such that *gcd*(*k*−1, 22−1) = 1 and *k*×22<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base22.txt, https://oeis.org/A273987)
* 182 is the smallest generalized Sierpinski number to base 23 (i.e. numbers *k* such that *gcd*(*k*+1, 23−1) = 1 and *k*×23<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base23.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 68×23<sup>365239</sup>+1)**
* 476 is the smallest generalized Riesel number to base 23 (i.e. numbers *k* such that *gcd*(*k*−1, 23−1) = 1 and *k*×23<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base23.txt, https://oeis.org/A273987)
* 30651 is the smallest generalized Sierpinski number to base 24 (i.e. numbers *k* such that *gcd*(*k*+1, 24−1) = 1 and *k*×24<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base24.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 32336 is the smallest non-(*m*<sup>2</sup> with *m* == 2, 3 mod 5 or 6×*m*<sup>2</sup> with *m* == 1, 4 mod 5) generalized Riesel number to base 24 (i.e. numbers *k* such that *gcd*(*k*−1, 24−1) = 1 and *k*×24<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base24.txt, https://oeis.org/A273987)
* 262638 is the smallest generalized Sierpinski number to base 25 (i.e. numbers *k* such that *gcd*(*k*+1, 25−1) = 1 and *k*×25<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base25.zip, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 346802 is the smallest non-square generalized Riesel number to base 25 (i.e. numbers *k* such that *gcd*(*k*−1, 25−1) = 1 and *k*×25<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base25.zip, https://oeis.org/A273987)
* 221 is the smallest generalized Sierpinski number to base 26 (i.e. numbers *k* such that *gcd*(*k*+1, 26−1) = 1 and *k*×26<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base26.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 149 is the smallest generalized Riesel number to base 26 (i.e. numbers *k* such that *gcd*(*k*−1, 26−1) = 1 and *k*×26<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base26.txt, https://oeis.org/A273987) **(solved, largest prime is 115×26<sup>520277</sup>−1)**
* 538 is the smallest non-cube generalized Sierpinski number to base 27 (i.e. numbers *k* such that *gcd*(*k*+1, 27−1) = 1 and *k*×27<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base27.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 804 is the smallest non-cube generalized Riesel number to base 27 (i.e. numbers *k* such that *gcd*(*k*−1, 27−1) = 1 and *k*×27<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base27.txt, https://oeis.org/A273987)
* 4554 is the smallest generalized Sierpinski number to base 28 (i.e. numbers *k* such that *gcd*(*k*+1, 28−1) = 1 and *k*×28<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base28.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 9078 is the smallest generalized Riesel number to base 28 (i.e. numbers *k* such that *gcd*(*k*−1, 28−1) = 1 and *k*×28<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base28.txt, https://oeis.org/A273987)
* 4 is the smallest generalized Sierpinski number to base 29 (i.e. numbers *k* such that *gcd*(*k*+1, 29−1) = 1 and *k*×29<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base29.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 2×29<sup>1</sup>+1)**
* 4 is the smallest generalized Riesel number to base 29 (i.e. numbers *k* such that *gcd*(*k*−1, 29−1) = 1 and *k*×29<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base29.txt, https://oeis.org/A273987) **(solved, largest prime is 2×29<sup>136</sup>−1)**
* 867 is the smallest generalized Sierpinski number to base 30 (i.e. numbers *k* such that *gcd*(*k*+1, 30−1) = 1 and *k*×30<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base30.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 4928 is the smallest generalized Riesel number to base 30 (i.e. numbers *k* such that *gcd*(*k*−1, 30−1) = 1 and *k*×30<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base30.txt, https://oeis.org/A273987) other than 1369
* 6360528 is the smallest generalized Sierpinski number to base 31 (i.e. numbers *k* such that *gcd*(*k*+1, 31−1) = 1 and *k*×31<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base31.zip, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 134718 is the smallest generalized Riesel number to base 31 (i.e. numbers *k* such that *gcd*(*k*−1, 31−1) = 1 and *k*×31<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base31.txt, https://oeis.org/A273987)
* 10 is the smallest non-5th-power generalized Sierpinski number to base 32 (i.e. numbers *k* such that *gcd*(*k*+1, 32−1) = 1 and *k*×32<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base32.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 10 is the smallest non-5th-power generalized Riesel number to base 32 (i.e. numbers *k* such that *gcd*(*k*−1, 32−1) = 1 and *k*×32<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base32.txt, https://oeis.org/A273987) **(solved, largest prime is 3×32<sup>11</sup>−1)**
* 1854 is the smallest generalized Sierpinski number to base 33 (i.e. numbers *k* such that *gcd*(*k*+1, 33−1) = 1 and *k*×33<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base33.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 766×33<sup>610412</sup>+1)**
* 764 is the smallest non-(*m*<sup>2</sup> with *m* == 4, 13 mod 17 or 33×*m*<sup>2</sup> with *m* == 4, 13 mod 17) generalized Riesel number to base 33 (i.e. numbers *k* such that *gcd*(*k*−1, 33−1) = 1 and *k*×33<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base33.txt, https://oeis.org/A273987) **(solved, largest prime is 732×33<sup>19011</sup>−1)**
* 6 is the smallest generalized Sierpinski number to base 34 (i.e. numbers *k* such that *gcd*(*k*+1, 34−1) = 1 and *k*×34<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base34.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159) **(solved, largest prime is 1×34<sup>4</sup>+1)**
* 6 is the smallest non-(*m*<sup>2</sup> with *m* == 2, 3 mod 5 or 34×*m*<sup>2</sup> with *m* == 2, 3 mod 5) generalized Riesel number to base 34 (i.e. numbers *k* such that *gcd*(*k*−1, 34−1) = 1 and *k*×34<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base34.txt, https://oeis.org/A273987) **(solved, largest prime is 5×34<sup>2</sup>−1)**
* 214018 is the smallest generalized Sierpinski number to base 35 (i.e. numbers *k* such that *gcd*(*k*+1, 35−1) = 1 and *k*×35<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base35.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 287860 is the smallest generalized Riesel number to base 35 (i.e. numbers *k* such that *gcd*(*k*−1, 35−1) = 1 and *k*×35<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base35.zip, https://oeis.org/A273987)
* 1886 is the smallest generalized Sierpinski number to base 36 (i.e. numbers *k* such that *gcd*(*k*+1, 36−1) = 1 and *k*×36<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base36.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
* 116364 is the smallest non-square generalized Riesel number to base 36 (i.e. numbers *k* such that *gcd*(*k*−1, 36−1) = 1 and *k*×36<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base36.txt, https://oeis.org/A273987)
* 78557 is the smallest dual Sierpinski number (i.e. odd numbers *k* such that 2<sup>*n*</sup>+*k* is composite for all *n* ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758) **(solved if we allow probable primes, largest (probable) prime is 2<sup>9092392</sup>+40291)**
* 509203 is the smallest dual Riesel number (i.e. odd numbers *k* such that 2<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 2<sup>*n*</sup> > *k*) (https://mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462)
* 271129 is the second-smallest dual Sierpinski number (i.e. odd numbers *k* such that 2<sup>*n*</sup>+*k* is composite for all *n* ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758)
* 762701 is the second-smallest dual Riesel number (i.e. odd numbers *k* such that 2<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 2<sup>*n*</sup> > *k*) (https://mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462)
* 125050976086 is the smallest generalized dual Sierpinski number to base 3 (i.e. numbers *k* such that gcd(*k*, 3) = 1 and *gcd*(*k*+1, 3−1) = 1 and 3<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 63064644938 is the smallest generalized dual Riesel number to base 3 (i.e. numbers *k* such that gcd(*k*, 3) = 1 and *gcd*(*k*−1, 3−1) = 1 and 3<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 3<sup>*n*</sup> > *k*)
* 159986 is the smallest generalized dual Sierpinski number to base 5 (i.e. numbers *k* such that gcd(*k*, 5) = 1 and *gcd*(*k*+1, 5−1) = 1 and 5<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 346802 is the smallest generalized dual Riesel number to base 5 (i.e. numbers *k* such that gcd(*k*, 5) = 1 and *gcd*(*k*−1, 5−1) = 1 and 5<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 5<sup>*n*</sup> > *k*)
* 1112646039348 is the smallest generalized dual Sierpinski number to base 7 (i.e. numbers *k* such that gcd(*k*, 7) = 1 and *gcd*(*k*+1, 7−1) = 1 and 7<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 408034255082 is the smallest generalized dual Riesel number to base 7 (i.e. numbers *k* such that gcd(*k*, 7) = 1 and *gcd*(*k*−1, 7−1) = 1 and 7<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 7<sup>*n*</sup> > *k*)
* 1490 is the smallest generalized dual Sierpinski number to base 11 (i.e. numbers *k* such that gcd(*k*, 11) = 1 and *gcd*(*k*+1, 11−1) = 1 and 11<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 862 is the smallest generalized dual Riesel number to base 11 (i.e. numbers *k* such that gcd(*k*, 11) = 1 and *gcd*(*k*−1, 11−1) = 1 and 11<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 11<sup>*n*</sup> > *k*)
* 132 is the smallest generalized dual Sierpinski number to base 13 (i.e. numbers *k* such that gcd(*k*, 13) = 1 and *gcd*(*k*+1, 13−1) = 1 and 13<sup>*n*</sup>+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 13<sup>416</sup>+120)**
* 302 is the smallest generalized dual Riesel number to base 13 (i.e. numbers *k* such that gcd(*k*, 13) = 1 and *gcd*(*k*−1, 13−1) = 1 and 13<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 13<sup>*n*</sup> > *k*)
* 278 is the smallest generalized dual Sierpinski number to base 17 (i.e. numbers *k* such that gcd(*k*, 17) = 1 and *gcd*(*k*+1, 17−1) = 1 and 17<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 86 is the smallest generalized dual Riesel number to base 17 (i.e. numbers *k* such that gcd(*k*, 17) = 1 and *gcd*(*k*−1, 17−1) = 1 and 17<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 17<sup>*n*</sup> > *k*) **(solved, largest prime is 17<sup>18</sup>−80)**
* 765174 is the smallest generalized dual Sierpinski number to base 19 (i.e. numbers *k* such that gcd(*k*, 19) = 1 and *gcd*(*k*+1, 19−1) = 1 and 19<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 1119866 is the smallest non-(*m*<sup>2</sup> with *m* == 2, 3 mod 5 or 19×*m*<sup>2</sup> with *m* == 2, 3 mod 13) generalized dual Riesel number to base 19 (i.e. numbers *k* such that gcd(*k*, 19) = 1 and *gcd*(*k*−1, 19−1) = 1 and 19<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 19<sup>*n*</sup> > *k*)
* 182 is the smallest generalized dual Sierpinski number to base 23 (i.e. numbers *k* such that gcd(*k*, 23) = 1 and *gcd*(*k*+1, 23−1) = 1 and 23<sup>*n*</sup>+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 23<sup>1926</sup>+82)**
* 476 is the smallest generalized dual Riesel number to base 23 (i.e. numbers *k* such that gcd(*k*, 23) = 1 and *gcd*(*k*−1, 23−1) = 1 and 23<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 23<sup>*n*</sup> > *k*)
* 4 is the smallest generalized dual Sierpinski number to base 29 (i.e. numbers *k* such that gcd(*k*, 29) = 1 and *gcd*(*k*+1, 29−1) = 1 and 29<sup>*n*</sup>+*k* is composite for all *n* ≥ 1) **(solved, largest prime is 29<sup>1</sup>+2)**
* 4 is the smallest generalized dual Riesel number to base 29 (i.e. numbers *k* such that gcd(*k*, 29) = 1 and *gcd*(*k*−1, 29−1) = 1 and 29<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 29<sup>*n*</sup> > *k*) **(solved, largest prime is 29<sup>2</sup>−2)**
* 6360528 is the smallest generalized dual Sierpinski number to base 31 (i.e. numbers *k* such that gcd(*k*, 31) = 1 and *gcd*(*k*+1, 31−1) = 1 and 31<sup>*n*</sup>+*k* is composite for all *n* ≥ 1)
* 134718 is the smallest generalized dual Riesel number to base 31 (i.e. numbers *k* such that gcd(*k*, 31) = 1 and *gcd*(*k*−1, 31−1) = 1 and 31<sup>*n*</sup>−*k* is composite for all *n* ≥ 1 such that 31<sup>*n*</sup> > *k*)
* 201446503145165177 is the smallest reverse Sierpinski base to *k* = 2 (i.e. bases *b* such that gcd(2+1, *b*−1) = 1 and 2×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://mersenneforum.org/showthread.php?t=6918, https://mersenneforum.org/showthread.php?t=19725, https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A119624, https://oeis.org/A253178, https://oeis.org/A098872)
* There are no reverse Riesel bases to *k* = 2 (i.e. bases *b* such that gcd(2−1, *b*−1) = 1 and 2×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591, https://oeis.org/A098873, https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* There are no reverse Sierpinski bases to *k* = 3 (i.e. bases *b* such that gcd(3+1, *b*−1) = 1 and 3×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098877)
* There are no reverse Riesel bases to *k* = 3 (i.e. bases *b* such that gcd(3−1, *b*−1) = 1 and 3×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098876)
* 14 is the smallest reverse Sierpinski base to *k* = 4 (i.e. bases *b* such that gcd(4+1, *b*−1) = 1 and 4×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 4×12<sup>2</sup>+1)**
* 14 is the smallest non-square reverse Riesel base to *k* = 4 (i.e. bases *b* such that gcd(4−1, *b*−1) = 1 and 4×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 4×12<sup>1</sup>−1)**
* 140324348 is the smallest reverse Sierpinski base to *k* = 5 (i.e. bases *b* such that gcd(5+1, *b*−1) = 1 and 5×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* There are no reverse Riesel bases to *k* = 5 (i.e. bases *b* such that gcd(5−1, *b*−1) = 1 and 5×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 34 is the smallest reverse Sierpinski base to *k* = 6 (i.e. bases *b* such that gcd(6+1, *b*−1) = 1 and 6×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 6×20<sup>15</sup>+1)**
* 34 is the smallest non-(6×*m*<sup>2</sup> with *m* == 2, 3 mod 5) reverse Riesel base to *k* = 6 (i.e. bases *b* such that gcd(6−1, *b*−1) = 1 and 6×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 6×27<sup>2</sup>−1)**
* There are no reverse Sierpinski bases to *k* = 7 (i.e. bases *b* such that gcd(7+1, *b*−1) = 1 and 7×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 9162668342 is the smallest reverse Riesel base to *k* = 7 (i.e. bases *b* such that gcd(7−1, *b*−1) = 1 and 7×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 20 is the smallest non-cube reverse Sierpinski base to *k* = 8 (i.e. bases *b* such that gcd(8+1, *b*−1) = 1 and 8×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 8×6<sup>4</sup>+1)**
* 20 is the smallest non-cube reverse Riesel base to *k* = 8 (i.e. bases *b* such that gcd(8−1, *b*−1) = 1 and 8×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 8×7<sup>4</sup>−1)**
* 177744 is the smallest reverse Sierpinski base to *k* = 9 (i.e. bases *b* such that gcd(9+1, *b*−1) = 1 and 9×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* There are no reverse Riesel bases to *k* = 9 (i.e. bases *b* such that gcd(9−1, *b*−1) = 1 and 9×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) except the even square bases *b* and the bases *b* == 4 mod 10
* 32 is the smallest reverse Sierpinski base to *k* = 10 (i.e. bases *b* such that gcd(10+1, *b*−1) = 1 and 10×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 10×17<sup>1356</sup>+1)**
* 32 is the smallest reverse Riesel base to *k* = 10 (i.e. bases *b* such that gcd(10−1, *b*−1) = 1 and 10×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 10×17<sup>117</sup>−1)**
* 14 is the smallest reverse Sierpinski base to *k* = 11 (i.e. bases *b* such that gcd(11+1, *b*−1) = 1 and 11×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 11×12<sup>3</sup>+1)**
* 14 is the smallest reverse Riesel base to *k* = 11 (i.e. bases *b* such that gcd(11−1, *b*−1) = 1 and 11×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 11×8<sup>18</sup>−1)**
* 142 is the smallest reverse Sierpinski base to *k* = 12 (i.e. bases *b* such that gcd(12+1, *b*−1) = 1 and 12×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 142 is the smallest reverse Riesel base to *k* = 12 (i.e. bases *b* such that gcd(12−1, *b*−1) = 1 and 12×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 12×98<sup>3599</sup>−1)**
* 20 is the smallest reverse Sierpinski base to *k* = 13 (i.e. bases *b* such that gcd(13+1, *b*−1) = 1 and 13×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 13×18<sup>10</sup>+1)**
* 20 is the smallest reverse Riesel base to *k* = 13 (i.e. bases *b* such that gcd(13−1, *b*−1) = 1 and 13×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 13×12<sup>2</sup>−1)**
* 38 is the smallest reverse Sierpinski base to *k* = 14 (i.e. bases *b* such that gcd(14+1, *b*−1) = 1 and 14×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 14×23<sup>5</sup>+1)**
* 8 is the smallest reverse Riesel base to *k* = 14 (i.e. bases *b* such that gcd(14−1, *b*−1) = 1 and 14×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) **(solved, largest prime is 14×5<sup>2</sup>−1)**
* There are no reverse Sierpinski bases to *k* = 15 (i.e. bases *b* such that gcd(15+1, *b*−1) = 1 and 15×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 8241218 is the smallest reverse Riesel base to *k* = 15 (i.e. bases *b* such that gcd(15−1, *b*−1) = 1 and 15×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 38 is the smallest reverse Sierpinski base to *k* = 16 (i.e. bases *b* such that gcd(16+1, *b*−1) = 1 and 16×*b*<sup>*n*</sup>+1 is composite for all *n* ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
* 50 is the smallest non-square reverse Riesel base to *k* = 16 (i.e. bases *b* such that gcd(16−1, *b*−1) = 1 and 16×*b*<sup>*n*</sup>−1 is composite for all *n* ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) other than 33 **(solved, largest prime is 16×39<sup>35</sup>−1)**
* 201446503145165177 is the smallest dual reverse Sierpinski base to *k* = 2 (i.e. bases *b* such that gcd(2, *b*) = 1 and gcd(2+1, *b*−1) = 1 and *b*<sup>*n*</sup>+2 is composite for all *n* ≥ 1) (https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)
* There are no dual reverse Riesel bases to *k* = 2 (i.e. bases *b* such that gcd(2, *b*) = 1 and gcd(2−1, *b*−1) = 1 and *b*<sup>*n*</sup>−2 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 2) (https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A255707, https://oeis.org/A084714, https://oeis.org/A250200, https://oeis.org/A292201)
* There are no dual reverse Sierpinski bases to *k* = 3 (i.e. bases *b* such that gcd(3, *b*) = 1 and gcd(3+1, *b*−1) = 1 and *b*<sup>*n*</sup>+3 is composite for all *n* ≥ 1)
* There are no dual reverse Riesel bases to *k* = 3 (i.e. bases *b* such that gcd(3, *b*) = 1 and gcd(3−1, *b*−1) = 1 and *b*<sup>*n*</sup>−3 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 3)
* 140324348 is the smallest dual reverse Sierpinski base to *k* = 5 (i.e. bases *b* such that gcd(5, *b*) = 1 and gcd(5+1, *b*−1) = 1 and *b*<sup>*n*</sup>+5 is composite for all *n* ≥ 1)
* There are no dual reverse Riesel bases to *k* = 5 (i.e. bases *b* such that gcd(5, *b*) = 1 and gcd(5−1, *b*−1) = 1 and *b*<sup>*n*</sup>−5 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 5)
* There are no dual reverse Sierpinski bases to *k* = 7 (i.e. bases *b* such that gcd(7, *b*) = 1 and gcd(7+1, *b*−1) = 1 and *b*<sup>*n*</sup>+7 is composite for all *n* ≥ 1)
* 9162668342 is the smallest dual reverse Riesel base to *k* = 7 (i.e. bases *b* such that gcd(7, *b*) = 1 and gcd(7−1, *b*−1) = 1 and *b*<sup>*n*</sup>−7 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 7)
* 14 is the smallest dual reverse Sierpinski base to *k* = 11 (i.e. bases *b* such that gcd(11, *b*) = 1 and gcd(11+1, *b*−1) = 1 and *b*<sup>*n*</sup>+11 is composite for all *n* ≥ 1) **(solved, largest prime is 12<sup>1</sup>+11)**
* 74 is the smallest dual reverse Riesel base to *k* = 11 (i.e. bases *b* such that gcd(11, *b*) = 1 and gcd(11−1, *b*−1) = 1 and *b*<sup>*n*</sup>−11 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 11) **(solved, largest prime is 68<sup>6</sup>−11)** (note that for *b* = 14, the *only one* prime of the form 14<sup>*n*</sup>−11 with *n* ≥ 1 is 14<sup>1</sup>−11 = 3)
* 20 is the smallest dual reverse Sierpinski base to *k* = 13 (i.e. bases *b* such that gcd(13, *b*) = 1 and gcd(13+1, *b*−1) = 1 and *b*<sup>*n*</sup>+13 is composite for all *n* ≥ 1) **(solved, largest prime is 14<sup>16</sup>+13)**
* 38 is the smallest dual reverse Riesel base to *k* = 13 (i.e. bases *b* such that gcd(13, *b*) = 1 and gcd(13−1, *b*−1) = 1 and *b*<sup>*n*</sup>−13 is composite for all *n* ≥ 1 such that *b*<sup>*n*</sup> > 13) **(solved, largest prime is 14<sup>3</sup>−13)** (note that for *b* = 20, the *only one* prime of the form 20<sup>*n*</sup>−13 with *n* ≥ 1 is 20<sup>1</sup>−13 = 7)

We call families of the form *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) "linear" families. Our
algorithm then proceeds as follows:

1. *M* := {minimal primes in base *b* of length 2 or 3}, *L* := union of all *x*{*Y*}*z* such that *x* ≠ 0 and *gcd*(*z*, *b*) = 1 and *Y* is the set of digits *y* such that *xyz* has no subsequence in *M*.
2. While *L* contains nonlinear families (families which are not linear families): Explore each family of *L*, and update *L*. Examine each family of *L* by:
2.1. Let *w* be the shortest string in the family. If *w* has a subsequence in *M*, then remove the family from *L*. If *w* represents a prime, then add *w* to *M* and remove the family from *L*.
2.2. If possible, simplify the family.
2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites, and if so then remove the family from *L*.
3. Update *L*, after each split examine the new families as in step 2.

e.g. in decimal (base *b* = 10):

*M* := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

*L* := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to *L*

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to *L*

and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed

etc.

The process of exploring/examining/splitting a family can be concisely expressed in a tree of decompositions.

We should first make data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ *b* ≤ 26), then searching each left linear family to certain limit of length (say length 1000) (just like the new base script for Conjectures ‘R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://t5k.org/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem in bases 2 ≤ *b* ≤ 28), then find the smallest prime in each left linear family (use *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve, then use *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) to test the probable-primality of the remain numbers, then use *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to prove the primality of the probable primes < 10<sup>25000</sup>).

Some *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families can be proven to contain no primes > *b*, by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://web.archive.org/web/20060925100410/http://www.glasgowg43.freeserve.co.uk/siernums.htm, https://web.archive.org/web/20061116164533/http://www.glasgowg43.freeserve.co.uk/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) *S* of primes *p* such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of *S* (this is equivalent to finding a positive integer *N* such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *N*, this *N* is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b*, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11<sup>*n*</sup>−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime *p* such that all numbers in a given family are not *p*-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>), or combine of them (https://mersenneforum.org/showthread.php?t=11143, https://mersenneforum.org/showthread.php?t=10279, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://oeis.org/A213353, https://oeis.org/A233469), for me, there is nothing worse then searching a family for a long time that later is proven to contain no primes (e.g. we searched the base 16 families {C}D and {C}DD to length 10000 and searched the base 21 family B0{H}6H to length 20000 and searched the base 30 family A{0}9J to length 10000 in the past, and later we proved that the base 16 families {C}D and {C}DD both have Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup> and the base 21 family B0{H}6H has a single trivial factor 4637 and the base 30 family A{0}9J has a covering set of {7, 13, 19, 31} with period 6, this is because the original program cannot remove the families which can be ruled out as only containing composites by these conditions: "Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup>" and "single trivial prime factor > *b*<sup>2</sup>" and "covering congruence with period > 4", now the program can remove the families which can be ruled out as only containing composites by these conditions: "single trivial prime factor < *b*<sup>3</sup>" and "covering congruence with period ≤ 24" and "difference of *r*-th powers with *r* ≤ 5" and "Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup>").

The multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A086145) is very important in this problem, since if a prime *p* divides the number with *n* digits in family *x*{*y*}*z* in base *b*, then *p* also divides the number with *k*×*ord*<sub>*p*</sub>(*b*)+*n* digits in family *x*{*y*}*z* in base *b* for all nonnegative integer *k* (unless *ord*<sub>*p*</sub>(*b*) = 1, i.e. *p* divides *b*−1, in this case *p* also divides the number with *k*×*p*+*n* digits in family *x*{*y*}*z* in base *b* for all nonnegative integer *k*), the period of "divisible by *p*" for a prime *p* in family *x*{*y*}*z* in base *b* (if only some and not all numbers in family *x*{*y*}*z* in base *b* are divisible by *p*, of course, if all numbers in family *x*{*y*}*z* in base *b* are divisible by *p*, then the period of "divisible by *p*" for a prime *p* in family *x*{*y*}*z* in base *b* is 1) is *ord*<sub>*p*</sub>(*b*) (*ord*<sub>*p*</sub>(*b*) must divide *p*−1, if and only if *ord*<sub>*p*</sub>(*b*) is exactly *p*−1, then *b* is a primitive root (https://en.wikipedia.org/wiki/Primitive_root_modulo_n, https://mathworld.wolfram.com/PrimitiveRoot.html, https://oeis.org/A060749, https://oeis.org/A001918, https://oeis.org/A071894, https://oeis.org/A008330, https://oeis.org/A046147, https://oeis.org/A046145, https://oeis.org/A046146, https://oeis.org/A046144, https://oeis.org/A033948, https://oeis.org/A033949, http://www.bluetulip.org/2014/programs/primitive.html, http://www.numbertheory.org/php/lprimroot.html) mod *p*, and this is studying in Artin's conjecture on primitive roots (https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots, https://mathworld.wolfram.com/ArtinsConjecture.html, http://www.numericana.com/answer/constants.htm#artin), which is an unsolved problem in mathematics) unless *p* divides *b*−1, in this case the period of "divisible by *p*" for such prime *p* in family *x*{*y*}*z* in base *b* is simply *p*, the primes *p* such that *ord*<sub>*p*</sub>(*b*) = *n* are exactly the prime factors of the Zsigmondy number (https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem, https://mathworld.wolfram.com/ZsigmondyTheorem.html) *Zs*(*n*, *b*, 1), *Zs*(*n*, *b*, 1) = *Φ*<sub>*n*</sub>(*b*)/*gcd*(*Φ*<sub>*n*</sub>(*b*), *n*) (where *Φ* is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) if *n* ≠ 2, *Zs*(*n*, 2, 1) = odd part (http://mathworld.wolfram.com/OddPart.html, https://oeis.org/A000265) of *n*+1, the prime factors of *Zs*(*n*, *b*, 1) for odd *n* are exactly the primitive prime factors of *b*<sup>*n*</sup>−1, the prime factors of *Zs*(*n*, *b*, 1) for even *n* are exactly the primitive prime factors of *b*<sup>*n*/2</sup>+1, references: https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119 (list of the *ord*<sub>*p*</sub>(*b*) for 2 ≤ *b* ≤ 128 and primes *p* ≤ 4096), https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339 (list of primes *p* such that *ord*<sub>*p*</sub>(*b*) = *n* for 2 ≤ *b* ≤ 64 and 1 ≤ *n* ≤ 64), also factorization of *b*<sup>*n*</sup>±1: https://homes.cerias.purdue.edu/~ssw/cun/index.html (2 ≤ *b* ≤ 12), https://homes.cerias.purdue.edu/~ssw/cun/pmain423.txt (2 ≤ *b* ≤ 12), https://doi.org/10.1090/conm/022 (2 ≤ *b* ≤ 12), https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf) (2 ≤ *b* ≤ 12), https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/ (2 ≤ *b* ≤ 12), https://maths-people.anu.edu.au/~brent/factors.html (13 ≤ *b* ≤ 99), https://stdkmd.net/nrr/repunit/ (*b* = 10), https://stdkmd.net/nrr/repunit/10001.htm (*b* = 10), https://stdkmd.net/nrr/repunit/phin10.htm (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.lz (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.gz (*b* = 10, only primitive factors), https://kurtbeschorner.de/ (*b* = 10), https://kurtbeschorner.de/fact-2500.htm (*b* = 10), https://repunit-koide.jimdofree.com/ (*b* = 10), https://repunit-koide.jimdofree.com/app/download/10034950550/Repunit100-20230630.pdf?t=1688135997 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_242.pdf) (*b* = 10), https://gmplib.org/~tege/repunit.html (*b* = 10), https://gmplib.org/~tege/fac10m.txt (*b* = 10), https://gmplib.org/~tege/fac10p.txt (*b* = 10), https://web.archive.org/web/20120426061657/http://oddperfect.org/ (prime *b*), http://myfactors.mooo.com/ (any *b*), http://myfactorcollection.mooo.com:8090/dbio.html (any *b*), http://www.asahi-net.or.jp/~KC2H-MSM/cn/old/index.htm (any *b*, only primitive factors), http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm (any *b*, only primitive factors), https://web.archive.org/web/20050922233702/http://user.ecc.u-tokyo.ac.jp/~g440622/cn/index.html (any *b*, only primitive factors), also for the factors of *b*<sup>*n*</sup>±1 with 2 ≤ *b* ≤ 100 and 1 ≤ *n* ≤ 100 see http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=1&TExp=100&c0=&LM= (only primitive factors); also, the period of "difference-of-two-squares factorization" in any base *b* is 2 if *b* is not square, 1 if *b* is square; the period of "sum/difference-of-two-*p*th-powers factorization with odd prime *p*" is *p* if *b* is not *p*-th power, 1 if *b* is *p*-th power; the period of "Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup>" is 4 if *b* is not square, 2 if *b* is square but not 4th power, 1 if *b* is 4th power, (for more information, see https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://oeis.org/A014664, https://oeis.org/A062117, https://oeis.org/A002371, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm) the family *x*{*y*}*z* in base *b* can be proven to contain no primes > *b* (or only contain finitely many primes > *b*) if and only if these residue classes with these periods gives a complete residue system (https://en.wikipedia.org/wiki/Covering_system, https://mathworld.wolfram.com/CompleteResidueSystem.html).

The above section only includes the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A086145) of the base (*b*) mod the primes (i.e. *ord*<sub>*p*</sub>(*b*) with prime *p*), if you want to calculate the multiplicative order of the base (*b*) mod a composite number *c* coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, factor (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) *c* to product of distinct prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), and calculate the multiplicative order of *b* mod *p*<sup>*e*</sup> (i.e. *ord*<sub>*p*<sup>*e*</sup></sub>(*b*)) for all these prime powers *p*<sup>*e*</sup>, and *ord*<sub>*p*<sup>*e*</sup></sub>(*b*) = *p*<sup>*max*(*e*−*r*(*b*,*p*),0)</sup>×*ord*<sub>*p*</sub>(*b*), where *r*(*b*,*p*) is the largest integer *s* such that *p*<sup>*s*</sup> divides *b*<sup>*p*−1</sup>−1, the primes *p* such that *r*(*b*,*p*) > 1 are called generalized Wieferich prime (https://en.wikipedia.org/wiki/Wieferich_prime, https://t5k.org/glossary/xpage/WieferichPrime.html, https://mathworld.wolfram.com/WieferichPrime.html, https://www.primegrid.com/stats_ww.php, https://oeis.org/A001220) base *b*.

This is a list for all known generalized Wieferich primes in bases 2 ≤ *b* ≤ 36 (*r*(*b*,*p*) = 2 for the generalized Wieferich primes *p* in base *b* with no orders listed (for these generalized Wieferich primes *p*, the orders are 1), and *r*(*b*,*p*) = (the order listed) + 1 for the generalized Wieferich primes *p* in base *b* with orders listed): (references: http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt, http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt, http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math11/fer_quo.htm, https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html (prime bases *b*), https://web.archive.org/web/20140810051836/http://www.cecm.sfu.ca/~mjm/WieferichBarker/WPBS1.html (prime bases *b*), http://www.bitman.name/math/table/489 (in Italian), http://www.urticator.net/essay/6/624.html, http://www.sci.kobe-u.ac.jp/old/seminar/pdf/2008_yamazaki.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_63.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1182246-5/S0025-5718-1993-1182246-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_185.pdf))

|*b*|generalized Wieferich primes base *b* (written in base 10)|search limit|*OEIS* sequence|
|---|---|---|---|
|2|1093, 3511, ...|6×10<sup>17</sup>|https://oeis.org/A001220|
|3|11, 1006003, ...|1.2×10<sup>15</sup>|https://oeis.org/A014127|
|4|1093, 3511, ...|6×10<sup>17</sup>|the same as https://oeis.org/A001220|
|5|2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...|1.2×10<sup>15</sup>|https://oeis.org/A123692|
|6|66161, 534851, 3152573, ...|1.63×10<sup>14</sup>|https://oeis.org/A212583|
|7|5, 491531, ...|1.2×10<sup>15</sup>|https://oeis.org/A123693|
|8|3, 1093, 3511, ...|6×10<sup>17</sup>|the same as https://oeis.org/A001220 plus the prime 3|
|9|2 (order 2), 11, 1006003, ...|1.2×10<sup>15</sup>|the same as https://oeis.org/A014127 plus the prime 2|
|10|3, 487, 56598313, ...|1.63×10<sup>14</sup>|https://oeis.org/A045616|
|11|71, ...|1.63×10<sup>14</sup>|–|
|12|2693, 123653, ...|1.63×10<sup>14</sup>|https://oeis.org/A111027|
|13|2, 863, 1747591, ...|1.63×10<sup>14</sup>|https://oeis.org/A128667|
|14|29, 353, 7596952219, ...|1.63×10<sup>14</sup>|https://oeis.org/A234810|
|15|29131, 119327070011, ...|1.63×10<sup>14</sup>|https://oeis.org/A242741|
|16|1093, 3511, ...|6×10<sup>17</sup>|the same as https://oeis.org/A001220|
|17|2 (order 3), 3, 46021, 48947, 478225523351, ...|1.63×10<sup>14</sup>|https://oeis.org/A128668|
|18|5, 7 (order 2), 37, 331, 33923, 1284043, ...|1.63×10<sup>14</sup>|https://oeis.org/A244260|
|19|3, 7 (order 2), 13, 43, 137, 63061489, ...|1.63×10<sup>14</sup>|https://oeis.org/A090968|
|20|281, 46457, 9377747, 122959073, ...|1.63×10<sup>14</sup>|https://oeis.org/A242982|
|21|2, ...|1.63×10<sup>14</sup>|–|
|22|13, 673, 1595813, 492366587, 9809862296159, ...|1.63×10<sup>14</sup>|https://oeis.org/A298951|
|23|13, 2481757, 13703077, 15546404183, 2549536629329, ...|1.63×10<sup>14</sup>|https://oeis.org/A128669|
|24|5, 25633, ...|1.63×10<sup>14</sup>|–|
|25|2 (order 2), 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...|1.2×10<sup>15</sup>|the same as https://oeis.org/A123692|
|26|3 (order 2), 5, 71, 486999673, 6695256707, ...|1.63×10<sup>14</sup>|https://oeis.org/A306255|
|27|11, 1006003, ...|1.2×10<sup>15</sup>|the same as https://oeis.org/A014127|
|28|3 (order 2), 19, 23, ...|1.63×10<sup>14</sup>|–|
|29|2, ...|1.63×10<sup>14</sup>|–|
|30|7, 160541, 94727075783, ...|1.63×10<sup>14</sup>|https://oeis.org/A306256|
|31|7, 79, 6451, 2806861, ...|1.63×10<sup>14</sup>|https://oeis.org/A331424|
|32|5, 1093, 3511, ...|6×10<sup>17</sup>|the same as https://oeis.org/A001220 plus the prime 5|
|33|2 (order 4), 233, 47441, 9639595369, ...|1.63×10<sup>14</sup>|–|
|34|46145917691, ...|1.63×10<sup>14</sup>|–|
|35|3, 1613, 3571, ...|1.63×10<sup>14</sup>|–|
|36|66161, 534851, 3152573, ...|1.63×10<sup>14</sup>|the same as https://oeis.org/A212583|

We can show that: For the case of covering congruence, the numbers in the family are not equal to any element in *S*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial; for the case of algebraic factorization (if the numbers are factored as *F* × *G* / *d*), both *F* and *G* are > *d*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial (the exceptions are the base 9 family {1} and the base 25 family {1} and the base 32 family {1}. For the base 9 family {1}, the algebraic form is (9<sup>*n*</sup>−1)/8 with *n* ≥ 2, and can be factored to (3<sup>*n*</sup>−1) × (3<sup>*n*</sup>+1) / 8, if *n* ≥ 3, then both 3<sup>*n*</sup>−1 and 3<sup>*n*</sup>+1 are > 8, thus these factorizations are nontrivial, it only remains to check the case *n* = 2, but the number with *n* = 2 is 10 = 2 × 5 is not prime; for the base 25 family {1}, the algebraic form is (25<sup>*n*</sup>−1)/24 with *n* ≥ 2, and can be factored to (5<sup>*n*</sup>−1) × (5<sup>*n*</sup>+1) / 24, if *n* ≥ 3, then both 5<sup>*n*</sup>−1 and 5<sup>*n*</sup>+1 are > 24, thus these factorizations are nontrivial, it only remains to check the case *n* = 2, but the number with *n* = 2 is 26 = 2 × 13 is not prime; for the base 32 family {1}, the algebraic form is (32<sup>*n*</sup>−1)/31 with *n* ≥ 2, and can be factored to (2<sup>*n*</sup>−1) × (16<sup>*n*</sup>+8<sup>*n*</sup>+4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 31, if *n* ≥ 6, then both 2<sup>*n*</sup>−1 and 16<sup>*n*</sup>+8<sup>*n*</sup>+4<sup>*n*</sup>+2<sup>*n*</sup>+1 are > 31, thus these factorizations are nontrivial, it only remains to check the cases *n* = 2, 3, 4, 5, but the numbers with *n* = 2, 3, 4, 5 are 33 = 3 × 11, 1057 = 7 × 151, 33825 = 3 × 5<sup>2</sup> × 11 × 41, 1082401 = 601 × 1801 are not primes); for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as *F* × *G* / *d*), the numbers in the family are not equal to any element in *S* and both *F* and *G* are > *d*, if *n* makes the numbers > *b*, thus these factorizations are nontrivial.

|type for proving the |possible bases *b*|such bases 2 ≤ *b* ≤ 36|
|---|---|---|
|covering congruence with 1 prime|any base *b*<br>(however, all such families in base *b* = 2 end with 0 and thus have trailing zeros (https://en.wikipedia.org/wiki/Trailing_zero) and thus not counted)|(2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36|
|covering congruence with 2 primes|*b* such that *b*+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)<br>(however, the case *b* = 5 has only four such families: {1}3, {1}4, 3{1}, 4{1}, all of them are covered by the prime 111 (31 in decimal), thus the smallest base *b* with families which have covering congruence with 2 primes is *b* = 9)|(5), 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35|
|covering congruence with 3 primes and period 3|*b* such that *omega*(*b*<sup>2</sup>+*b*+1) ≥ 3 (where *omega* is the omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221), the number of *distinct* primes dividing *n*)|16, 25|
|covering congruence with 3 primes and period 4|*b* such that *b*+1 is not a power of 2 (https://oeis.org/A000079, https://en.wikipedia.org/wiki/Power_of_two) and *b*<sup>2</sup>+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)<br>(however, bases *b* = 5, 8, 9, 11, 12, 18 has no such families, base 8 family 6{4}7 is covered by the prime 4<sub>220</sub>7)|(5), (8), (9), (11), (12), 13, 17, (18), 19, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 34, 35|
|algebraic factorization with difference of two squares|*b* such that *b* is square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html)<br>(however, base *b* = 4 has no such families, the family {1} has the prime 11 (5 in decimal))|(4), 9, 16, 25, 36|
|algebraic factorization with difference of two cubes|*b* such that *b* is cube (https://oeis.org/A000578, https://en.wikipedia.org/wiki/Cube_(algebra), https://mathworld.wolfram.com/CubicNumber.html)|8, 27|
|algebraic factorization with difference of two 5th powers|*b* such that *b* is 5th power (https://oeis.org/A000584, https://en.wikipedia.org/wiki/Fifth_power_(algebra))|32|
|algebraic factorization with *x*<sup>4</sup>+4×*y*<sup>4</sup>|*b* such that *b* is 4th power (https://oeis.org/A000583, https://en.wikipedia.org/wiki/Fourth_power, https://mathworld.wolfram.com/BiquadraticNumber.html)|16|
|combine of covering congruence with 1 prime and algebraic factorization with difference of two squares|*b* such that *b* is not square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html) and *b*+1 has a prime factor *p* == 1 mod 4|12, 14, 19, 24, 28, 29, 33, 34|

(You can see the factorization (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) of the numbers in these families in *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database), you have to convert them to algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form, and you will find that *all* numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) either "FF" or "CF", and no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (i.e. in http://factordb.com/listtype.php?t=3) (i.e. no numbers in these families are "genuine composites", i.e. compositeness is proved but no proper (prime or composite) factor is yet known) (of course, also no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", "P", "PRP" (i.e. in http://factordb.com/listtype.php?t=2, http://factordb.com/listtype.php?t=4, http://factordb.com/listtype.php?t=1), since all numbers in these families are known to be composite), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) with *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*<sup>*n*</sup>+*c* has algebraic factors, and their Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=421186&postcount=19, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) is zero, e.g. for the family 3{0}95 in base 13, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 3×13<sup>*n*+2</sup>+122, and in *factordb* you will find that all numbers in this family are divisible by some element of {5,7,17}, see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family {7}D in base 21, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (7×21<sup>*n*+1</sup>+113)/20, and in *factordb* you will find that all numbers in this family are divisible by some element of {2,13,17}, see http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 30{F}A0F in base 16, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 49×16<sup>*n*+3</sup>−1521, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family 5{1} in base 25, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×25<sup>*n*</sup>−1)/24, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family {D}5 in base 14, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 14<sup>*n*+1</sup>−9, and in *factordb* you will find that all numbers with even *n* in this family are divisible by 5, and you will find that no numbers with odd *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with odd *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 7{9} in base 17, its algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×17<sup>*n*</sup>−9)/16, and in *factordb* you will find that all numbers with odd *n* in this family are divisible by 2, and you will find that no numbers with even *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with even *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base). In contrast, you can see the factorization of the numbers in unsolved families in base *b* (which are listed in the "left *b*" file) in *factordb*, you will find some numbers in these families which have neither small prime factors (say < 10<sup>16</sup>) nor two nearly equal (prime or composite) factors, also you will find some numbers in these families which have no known proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1 (i.e. you will find some numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (instead of "CF" or "FF") (i.e. in http://factordb.com/listtype.php?t=3) (and maybe some large numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", i.e. in http://factordb.com/listtype.php?t=2) in *factordb* (http://factordb.com/), i.e. some numbers in these families are "genuine composites" (i.e. compositeness of these numbers are proved but no proper (prime or composite) factors of them are yet known)), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) with *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) even if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*<sup>*n*</sup>+*c* has algebraic factors, and they have positive Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=421186&postcount=19, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), and they have prime candidates (the Nash weight (or difficulty) tells you how many candidates remain after sieving a certain number of terms to a certain depth (say 10<sup>9</sup>), if the sieving program (i.e. *SRSIEVE*) was updated so that it also removes the *n* such that *a*×*b*<sup>*n*</sup>+*c* has algebraic factors), we can use the sense of http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://mersenneforum.org/showpost.php?p=138737&postcount=24, https://mersenneforum.org/showpost.php?p=153508&postcount=147, to show this, e.g. for the unsolved family A{3}A in base *b* = 13 (its algebraic form is (41×13<sup>*n*+1</sup>+27)/4, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show): )

* The algebraic form of A3<sub>*n*</sub>A (in base *b* = 13) is (41×13<sup>*n*+1</sup>+27)/4, and there is no *n* such that 41×13<sup>*n*+1</sup> is perfect power (after all, 41×13<sup>*n*+1</sup> is divisible by 41 but not 41<sup>2</sup>), thus the family A3<sub>*n*</sub>A (in base *b* = 13) has no algebraic factorization
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 2 if and only if *n* == 0 mod 2
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 17 if and only if *n* == 3 mod 4
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 37 if and only if *n* == 1 mod 36
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 47 if and only if *n* == 1 mod 46
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 11 if and only if *n* == 5 mod 10
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 53 if and only if *n* == 9 mod 13
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 31 if and only if *n* == 13 mod 30
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 75347 if and only if *n* == 17 mod 37673
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 71 if and only if *n* == 21 mod 70
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 167 if and only if *n* == 29 mod 166
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 523 if and only if *n* == 29 mod 261
* A3<sub>*n*</sub>A (in base *b* = 13) is divisible by 1321 if and only if *n* == 33 mod 1320

etc.

and it does not appear to be any covering set (of either primes or algebraic factors, or both), thus this form cannot be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and its Nash weight (or difficulty) is positive, and it has prime candidate, and hence there must be a prime at some point.

(for the examples of nonlinear families, see https://stdkmd.net/nrr/prime/primecount3.htm and https://stdkmd.net/nrr/prime/primecount3.txt (only base 10 families), nonlinear families usually have small primes if they cannot be ruled out as only containing composites by covering congruence, see the section above)

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and http://alfredreichlg.de/, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm)

e.g. (only list the families which all numbers do not contain "prime > *b*" subsequence) (see post https://mersenneforum.org/showpost.php?p=594923&postcount=231 for the factor pattern for some of these families) (for the case of covering congruence, we can show that the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, thus these factorizations are nontrivial; and for the case of algebraic factorization, we can show that both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial; for the case of combine of them, we can show that for the part of covering congruence, the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, and for the part of algebraic factorization, both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial)

(only list the families which can be proven to be not covered by any prime > *b* (this would include all such families of the form *x*{0}*y* and all such families of the form *x*{*y*} (unless *y* = 1) and all such families of the form {*x*}*y* (unless *x* = 1)), the only exceptions are the base 21 family B0{H}6H (which is covered by the smallest prime in the family B{H}6H in base 21 (if such prime exists)) and the base 8 family 6{4}7 (which is covered by the prime 4<sub>220</sub>7 in base 8))

(all small prime factors (< 10<sup>9</sup>, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in these families were added to *factordb*)

(this table is sorted by: single trivial prime factor (2{0}1 in base *b* = 10 through D{6}R in base *b* = 28) → covering congruence with 2 primes ({1}5 in base *b* = 9 through {X}5 in base *b* = 34) → covering congruence with ≥ 3 primes (6{4}7 in base *b* = 8 through {G}L in base *b* = 32) → algebraic factorization ({1} in base *b* = 9 through 9{S}IJ in base *b* = 36) → combine of covering congruence and algebraic factorization (8{D} in base *b* = 14 through {X}P in base *b* = 34))

|*b*<br>(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*d*) form of this family (*n* is the number of digits in the "{}", also the lower bound of *n* to make the numbers > *b*)<br>(note: *d* divides *gcd*(*a*+*c*,*b*−1), but *d* need not be *gcd*(*a*+*c*,*b*−1), *d* = *gcd*(*a*+*c*,*b*−1) if and only if the numbers in the family are not divisible by some prime factor of *b*−1, i.e. the numbers in the family are coprime to *b*−1, in fact, *d* = *gcd*(*a*+*c*,*b*−1)/*gcd*(*a*+*c*,*b*−1,(largest trivial factor of the family)))|why this family contain no primes > *b*|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))<br>(only for linear families)|
|---|---|---|---|---|
|10|2{0}1|2×10<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|2{0}7|2×10<sup>*n*+1</sup>+7 (*n* ≥ 0)|always divisible by 3<br>(in fact, always divisible by 9)|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|5{0}1|5×10<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|5{0}7|5×10<sup>*n*+1</sup>+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|8{0}1|8×10<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 3<br>(in fact, always divisible by 9)|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|8{0}7|8×10<sup>*n*+1</sup>+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|28{0}7|28×10<sup>*n*+1</sup>+7 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|4{6}9|(14×10<sup>*n*+1</sup>+7)/3 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|10|families ending with 0, 2, 4, 6, or 8|–|always divisible by 2|–|
|10|families ending with 0 or 5|–|always divisible by 5|–|
|10|{0,3,6,9}|–|always divisible by 3<br>(nonlinear family)|–|
|10|{0,7}|–|always divisible by 7<br>(nonlinear family)|–|
|any base (*b*)|families ending with digits *d* which are not coprime to *b*|–|always divisible by *gcd*(*d*,*b*)|–|
|any base (*b*)|families whose digits all have a common factor *d* > 1|–|always divisible by *d*|–|
|3|1{0}1|3<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|4|2{0}1|2×4<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*4%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|1{0}1|5<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|1{0}3|5<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 2<br>(in fact, always divisible by 4)|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|3{0}1|3×5<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2<br>(in fact, always divisible by 4)|http://factordb.com/index.php?query=3*5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|11{0}3|6×5<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|3{0}11|3×5<sup>*n*+2</sup>+6 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=3*5%5E%28n%2B2%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|6|4{0}1|4×6<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 5|http://factordb.com/index.php?query=4*6%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1{0}1|7<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1{0}3|7<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1{0}5|7<sup>*n*+1</sup>+5 (*n* ≥ 0)|always divisible by 2<br>(in fact, always divisible by 6)|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|3{0}1|3×7<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|3{0}5|3×7<sup>*n*+1</sup>+5 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|5{0}1|5×7<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 2<br>(in fact, always divisible by 6)|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|5{0}3|5×7<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1{0}2|7<sup>*n*+1</sup>+2 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|2{0}1|2×7<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|4{0}5|4×7<sup>*n*+1</sup>+5 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=4*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|5{0}4|5×7<sup>*n*+1</sup>+4 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|7|1{0}1{0}1|7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+1 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|1{0}3{0}5|7<sup>*m*+*n*+2</sup>+3×7<sup>*n*+1</sup>+5 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|1{0}5{0}3|7<sup>*m*+*n*+2</sup>+5×7<sup>*n*+1</sup>+3 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|3{0}1{0}5|3×7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+5 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|3{0}5{0}1|3×7<sup>*m*+*n*+2</sup>+5×7<sup>*n*+1</sup>+1 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|5{0}1{0}3|5×7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+3 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|5{0}3{0}1|5×7<sup>*m*+*n*+2</sup>+3×7<sup>*n*+1</sup>+1 (*m*+*n* ≥ 0)|always divisible by 3<br>(nonlinear family)|–|
|7|1{0}1{0}1{0}1|7<sup>*r*+*m*+*n*+3</sup>+7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+1 (*r*+*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|1{0}1{0}2|7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+2 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|1{0}2{0}1|7<sup>*m*+*n*+2</sup>+2×7<sup>*n*+1</sup>+1 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|2{0}1{0}1|2×7<sup>*m*+*n*+2</sup>+7<sup>*n*+1</sup>+1 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|4{0}5{0}5|4×7<sup>*m*+*n*+2</sup>+5×7<sup>*n*+1</sup>+5 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|5{0}4{0}5|5×7<sup>*m*+*n*+2</sup>+4×7<sup>*n*+1</sup>+5 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|7|5{0}5{0}4|5×7<sup>*m*+*n*+2</sup>+5×7<sup>*n*+1</sup>+4 (*m*+*n* ≥ 0)|always divisible by 2<br>(nonlinear family)|–|
|8|2{0}5|2×8<sup>*n*+1</sup>+5 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=2*8%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|4{0}3|4×8<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=4*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|6{0}1|6×8<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=6*8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|44{0}3|36×8<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=36*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|6{0}11|6×8<sup>*n*+2</sup>+9 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|3{4}1|(25×8<sup>*n*+1</sup>−25)/7 (*n* ≥ 0)|always divisible by 5<br>(in fact, always divisible by 25)<br>(in fact, also difference-of-two-squares factorization)<br>(25×8<sup>*n*+1</sup>−25)/7 = 25 × (2<sup>*n*</sup>−1) × (4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 7<br>(special example, as the numbers with length ≥ 10 in this family contain "prime > *b*" subsequence, this prime is 4<sub>8</sub>1)|http://factordb.com/index.php?query=%2825*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{7}62|(7×9<sup>*n*+2</sup>−119)/8 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-119%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|2{5}3|(5×11<sup>*n*+1</sup>−5)/2 (*n* ≥ 0)|always divisible by 5<br>(in fact, always divisible by 25)|http://factordb.com/index.php?query=%285*11%5E%28n%2B1%29-5%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{5}2|(7×11<sup>*n*+1</sup>−7)/2 (*n* ≥ 0)|always divisible by 5<br>(in fact, always divisible by 35)|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{5}9|(7×11<sup>*n*+1</sup>+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|A{5}2|(21×11<sup>*n*+1</sup>−7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|A{5}9|(21×11<sup>*n*+1</sup>+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|12|A{0}21|10×12<sup>*n*+2</sup>+25|always divisible by 5|http://factordb.com/index.php?query=10*12%5E%28n%2B2%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3{6}A|(7×13<sup>*n*+1</sup>+7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|5{A}C|(35×13<sup>*n*+1</sup>+7)/6 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2835*13%5E%28n%2B1%29%2B7%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|A{6}3|(21×13<sup>*n*+1</sup>−7)/2 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2821*13%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|C{A}5|(77×13<sup>*n*+1</sup>−35)/6 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2877*13%5E%28n%2B1%29-35%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|8{6}9|(110×14<sup>*n*+1</sup>+33)/13 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%28110*14%5E%28n%2B1%29%2B33%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|40{4}9|(732×14<sup>*n*+1</sup>+61)/13 (*n* ≥ 0)|always divisible by 61|http://factordb.com/index.php?query=%28732*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|15|9{6}8|(66×15<sup>*n*+1</sup>+11)/7 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%2866*15%5E%28n%2B1%29%2B11%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2{C}3|(14×16<sup>*n*+1</sup>−49)/5 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2814*16%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|A{4}1|(154×16<sup>*n*+1</sup>−49)/15 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%28154*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|8{C}F|(44×16<sup>*n*+1</sup>+11)/5 (*n* ≥ 0)|always divisible by 11<br>(in fact, also Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup><br>(44×16<sup>*n*+1</sup>+11)/5 = 11 × (2×4<sup>*n*+1</sup>−2×2<sup>*n*+1</sup>+1) × (2×4<sup>*n*+1</sup>+2×2<sup>*n*+1</sup>+1) / 5|http://factordb.com/index.php?query=%2844*16%5E%28n%2B1%29%2B11%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|8{A}F|(26×16<sup>*n*+1</sup>+13)/3 (*n* ≥ 0)|always divisible by 13|http://factordb.com/index.php?query=%2826*16%5E%28n%2B1%29%2B13%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|A{6}9|(52×16<sup>*n*+1</sup>+13)/5 (*n* ≥ 0)|always divisible by 13<br>(in fact, also Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup><br>(52×16<sup>*n*+1</sup>+13)/5 = 13 × (2×4<sup>*n*+1</sup>−2×2<sup>*n*+1</sup>+1) × (2×4<sup>*n*+1</sup>+2×2<sup>*n*+1</sup>+1) / 5|http://factordb.com/index.php?query=%2852*16%5E%28n%2B1%29%2B13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|21|B0{H}6H|(4637×21<sup>*n*+2</sup>−4637)/20 (*n* ≥ 0)|always divisible by 4637|http://factordb.com/index.php?query=%284637*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|4{O}9|(44×28<sup>*n*+1</sup>−143)/9 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%2844*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|N{6}R|(209×28<sup>*n*+1</sup>+187)/9 (*n* ≥ 0)|always divisible by 11|http://factordb.com/index.php?query=%28209*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|D{6}R|(119×28<sup>*n*+1</sup>+187)/9 (*n* ≥ 0)|always divisible by 17|http://factordb.com/index.php?query=%28119*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{1}5|(9<sup>*n*+1</sup>+31)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B1%29%2B31%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{1}61|(9<sup>*n*+2</sup>+359)/8 (*n* ≥ 0)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|16{1}|(121×9<sup>*n*</sup>−1)/8 (*n* ≥ 0)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even<br>(in fact, also difference-of-two-squares factorization)<br>(121×9<sup>*n*</sup>−1)/8 = (11×3<sup>*n*</sup>−1) × (11×3<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%28121*9%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|2{7}|(23×9<sup>*n*</sup>−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2823*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{3}5|(3×9<sup>*n*+1</sup>+13)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B13%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{3}8|(3×9<sup>*n*+1</sup>+37)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B37%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{3}05|(3×9<sup>*n*+2</sup>−203)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B2%29-203%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|5{1}|(41×9<sup>*n*</sup>−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|5{7}|(47×9<sup>*n*</sup>−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2847*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|6{1}|(49×9<sup>*n*</sup>−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is even, divisible by 5 if *n* is odd<br>(in fact, also difference-of-two-squares factorization)<br>(49×9<sup>*n*</sup>−1)/8 = (7×3<sup>*n*</sup>−1) × (7×3<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{7}2|(7×9<sup>*n*+1</sup>−47)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-47%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{7}5|(7×9<sup>*n*+1</sup>−23)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-23%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{7}05|(3×9<sup>*n*+2</sup>−527)/8 (*n* ≥ 1)|always divisible by some element of {2,5}<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-527%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{1}6{1}|(9<sup>*m*+*n*+1</sup>+40×9<sup>*n*</sup>−1)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *m*+*n* is even, divisible by 5 if *m*+*n* is odd|–|
|9|{7}2{7}|(7×9<sup>*m*+*n*+1</sup>−40×9<sup>*n*</sup>−7)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *m*+*n* is even, divisible by 5 if *m*+*n* is odd|–|
|9|5{0}{1}|(40×9<sup>*m*+*n*</sup>+9<sup>*n*</sup>−1)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|–|
|9|5{0}{7}|(40×9<sup>*m*+*n*</sup>+7×9<sup>*n*</sup>−7)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *n* is odd, divisible by 5 if *n* is even|–|
|9|{1}{0}5|(9<sup>*m*+*n*+1</sup>−9<sup>*n*+1</sup>+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–|
|9|{3}{0}5|(3×9<sup>*m*+*n*+1</sup>−3×9<sup>*n*+1</sup>+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–|
|9|{7}{0}5|(7×9<sup>*m*+*n*+1</sup>−7×9<sup>*n*+1</sup>+40)/8 (*m*+*n* ≥ 1)|always divisible by some element of {2,5}<br>(nonlinear family)<br>divisible by 2 if *m* is odd, divisible by 5 if *m* is even|–|
|11|2{5}|(5×11<sup>*n*</sup>−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{1}|(31×11<sup>*n*</sup>−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2831*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{5}|(7×11<sup>*n*</sup>−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{7}|(37×11<sup>*n*</sup>−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2837*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|4{1}|(41×11<sup>*n*</sup>−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2841*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|4{7}|(47×11<sup>*n*</sup>−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2847*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|8{5}|(17×11<sup>*n*</sup>−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2817*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|9{1}|(91×11<sup>*n*</sup>−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2891*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|9{5}|(19×11<sup>*n*</sup>−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2819*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|9{7}|(97×11<sup>*n*</sup>−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2897*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|A{1}|(101×11<sup>*n*</sup>−1)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%28101*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|A{7}|(107×11<sup>*n*</sup>−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%28107*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{5}2|(11<sup>*n*+1</sup>−7)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{1}3|(11<sup>*n*+1</sup>+19)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B19%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{5}3|(11<sup>*n*+1</sup>−5)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{7}3|(7×11<sup>*n*+1</sup>−47)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-47%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{1}4|(11<sup>*n*+1</sup>+29)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B29%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{7}4|(7×11<sup>*n*+1</sup>−37)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-37%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{5}8|(11<sup>*n*+1</sup>+5)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{1}9|(11<sup>*n*+1</sup>+79)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B79%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{5}9|(11<sup>*n*+1</sup>+7)/2 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{7}9|(7×11<sup>*n*+1</sup>+13)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B13%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{1}A|(11<sup>*n*+1</sup>+89)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if the length is odd, divisible by 3 if the length is even<br>(special example, as the numbers with length ≥ 18 in this family contain "prime > *b*" subsequence, this prime is 1<sub>17</sub>)|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B89%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|{7}A|(7×11<sup>*n*+1</sup>+23)/10 (*n* ≥ 1)|always divisible by some element of {2,3}<br>divisible by 2 if the length is odd, divisible by 3 if the length is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B23%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{0}{5}|(6×11<sup>*m*+*n*</sup>+11<sup>*n*</sup>−1)/2|always divisible by some element of {2,3}<br>(nonlinear family)<br>divisible by 2 if *n* is odd, divisible by 3 if *n* is even|–|
|11|{5}{0}3|(11<sup>*m*+*n*+1</sup>−11<sup>*n*+1</sup>+6)/2|always divisible by some element of {2,3}<br>(nonlinear family)<br>divisible by 2 if *m* is odd, divisible by 3 if *m* is even|–|
|14|4{0}1|4×14<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|B{0}1|11×14<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|3{D}|4×14<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=4*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|A{D}|11×14<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=11*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|1{0}B|14<sup>*n*+1</sup>+11 (*n* ≥ 0)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=14%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|{D}3|14<sup>*n*+1</sup>−11 (*n* ≥ 1)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=14%5E%28n%2B1%29-11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|{4}9|(4×14<sup>*n*+1</sup>+61)/13 (*n* ≥ 1)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%284*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|{8}5|(8×14<sup>*n*+1</sup>−47)/13 (*n* ≥ 1)|always divisible by some element of {3,5}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29-47%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|8{0}1|8×20<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=8*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|D{0}1|13×20<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=13*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|7{J}|8×20<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=8*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|C{J}|13×20<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=13*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|1{0}D|20<sup>*n*+1</sup>+13 (*n* ≥ 0)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|{J}7|20<sup>*n*+1</sup>−13 (*n* ≥ 1)|always divisible by some element of {3,7}<br>divisible by 3 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=20%5E%28n%2B1%29-13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|D{1}|(313×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|always divisible by some element of {2,13}<br>divisible by 2 if *n* is odd, divisible by 13 if *n* is even|http://factordb.com/index.php?query=%28313*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|E{1}|(337×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|always divisible by some element of {2,13}<br>divisible by 2 if *n* is even, divisible by 13 if *n* is odd|http://factordb.com/index.php?query=%28337*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|1E{1}|(937×25<sup>*n*</sup>−1)/24 (*n* ≥ 0)|always divisible by some element of {2,13}<br>divisible by 2 if *n* is odd, divisible by 13 if *n* is even|http://factordb.com/index.php?query=%28937*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|1F{1}|(961×25<sup>*n*</sup>−1)/24 (*n* ≥ 0)|always divisible by some element of {2,13}<br>divisible by 2 if *n* is even, divisible by 13 if *n* is odd<br>(in fact, also difference-of-two-squares factorization)<br>(961×25<sup>*n*</sup>−1)/24 = (31×5<sup>*n*</sup>−1) × (31×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28961*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|A{0}1|10×32<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=10*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|N{0}1|23×32<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=23*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|9{V}|10×32<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=10*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|M{V}|23×32<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=23*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|1{0}N|32<sup>*n*+1</sup>+23 (*n* ≥ 0)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is odd, divisible by 11 if *n* is even|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|{V}9|32<sup>*n*+1</sup>−23 (*n* ≥ 1)|always divisible by some element of {3,11}<br>divisible by 3 if *n* is even, divisible by 11 if *n* is odd|http://factordb.com/index.php?query=32%5E%28n%2B1%29-23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|6{0}1|6×34<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {5,7}<br>divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=6*34%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|5{X}|6×34<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {5,7}<br>divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=6*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|S{X}|29×34<sup>*n*</sup>−1 (*n* ≥ 1)|always divisible by some element of {5,7}<br>divisible by 5 if *n* is odd, divisible by 7 if *n* is even|http://factordb.com/index.php?query=29*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|{X}5|34<sup>*n*+1</sup>−29 (*n* ≥ 1)|always divisible by some element of {5,7}<br>divisible by 5 if *n* is even, divisible by 7 if *n* is odd|http://factordb.com/index.php?query=34%5E%28n%2B1%29-29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|6{4}7|(46×8<sup>*n*+1</sup>+17)/7 (*n* ≥ 0)|always divisible by some element of {3,5,13}<br>divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 13 if the length is == 0 mod 4<br>(special example, as the numbers with length ≥ 222 in this family contain "prime > *b*" subsequence, this prime is 4<sub>220</sub>7)|http://factordb.com/index.php?query=%2846*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|95{0}3|122×13<sup>*n*+1</sup>+3 (*n* ≥ 0)|always divisible by some element of {5,7,17}<br>divisible by 7 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 17 if *n* == 1 mod 4|http://factordb.com/index.php?query=122*13%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3{0}95|3×13<sup>*n*+2</sup>+122 (*n* ≥ 0)|always divisible by some element of {5,7,17}<br>divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 17 if *n* == 0 mod 4|http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{4}D|(4×16<sup>*n*+1</sup>+131)/15 (*n* ≥ 1)|always divisible by some element of {3,7,13}<br>divisible by 3 if *n* == 2 mod 3, divisible by 7 if *n* == 1 mod 3, divisible by 13 if *n* == 0 mod 3|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{8}F|(8×16<sup>*n*+1</sup>+97)/15 (*n* ≥ 1)|always divisible by some element of {3,7,13}<br>divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 13 if *n* == 1 mod 3|http://factordb.com/index.php?query=%288*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|7F{0}D|134×17<sup>*n*+1</sup>+13 (*n* ≥ 0)|always divisible by some element of {3,5,29}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 29 if *n* == 0 mod 4|http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|D{0}7F|13×17<sup>*n*+2</sup>+134 (*n* ≥ 0)|always divisible by some element of {3,5,29}<br>divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 29 if *n* == 1 mod 4|http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B134&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|21|{7}D|(7×21<sup>*n*+1</sup>+113)/20 (*n* ≥ 1)|always divisible by some element of {2,13,17}<br>divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 17 if *n* == 2 mod 4|http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|7L{0}1|182×23<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {3,5,53}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 53 if *n* == 0 mod 4|http://factordb.com/index.php?query=182*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|1{0}7L|23<sup>*n*+2</sup>+182 (*n* ≥ 0)|always divisible by some element of {3,5,53}<br>divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 53 if *n* == 1 mod 4|http://factordb.com/index.php?query=23%5E%28n%2B2%29%2B182&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|{D}GA|(13×23<sup>*n*+2</sup>+1439)/22 (*n* ≥ 0)|always divisible by some element of {2,5,7,37,79}<br>divisible by 2 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 37 if *n* == 7 mod 12, divisible by 79 if *n* == 2 mod 3|http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|L{5}L|(467×23<sup>*n*+1</sup>+347)/22 (*n* ≥ 0)|always divisible by some element of {2,5,7,13,37}<br>divisible by 2 if *n* is even, divisible by 5 if *n* == 1 mod 4, divisible by 7 if *n* == 0 mod 3, divisible by 13 if *n* == 1 mod 6, divisible by 37 if *n* == 11 mod 12|http://factordb.com/index.php?query=%28467*23%5E%28n%2B1%29%2B347%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|9{N}|(239×25<sup>*n*</sup>−23)/24 (*n* ≥ 1)|always divisible by some element of {3,7,31}<br>divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 31 if *n* == 1 mod 3|http://factordb.com/index.php?query=%28239*25%5En-23%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{N}E|(23×25<sup>*n*+1</sup>−239)/24 (*n* ≥ 1)|always divisible by some element of {3,7,31}<br>divisible by 3 if *n* == 2 mod 3, divisible by 7 if *n* == 0 mod 3, divisible by 31 if *n* == 1 mod 3|http://factordb.com/index.php?query=%2823*25%5E%28n%2B1%29-239%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|27{1}|(1369×25<sup>*n*</sup>−1)/24 (*n* ≥ 0)|always divisible by some element of {3,7,31}<br>divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 2 mod 3, divisible by 31 if *n* == 1 mod 3<br>(in fact, also difference-of-two-squares factorization)<br>(1369×25<sup>*n*</sup>−1)/24 = (37×5<sup>*n*</sup>−1) × (37×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%281369*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|JP{0}1|538×27<sup>*n*+1</sup>+1 (*n* ≥ 0)|always divisible by some element of {5,7,73}<br>divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 73 if *n* == 0 mod 4|http://factordb.com/index.php?query=538*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|1{0}JP|27<sup>*n*+2</sup>+538 (*n* ≥ 0)|always divisible by some element of {5,7,73}<br>divisible by 7 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 73 if *n* == 1 mod 4|http://factordb.com/index.php?query=27%5E%28n%2B2%29%2B538&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|J{0}2|19×27<sup>*n*+1</sup>+2 (*n* ≥ 0)|always divisible by some element of {5,7,73}<br>divisible by 7 if *n* is odd, divisible by 5 if *n* == 0 mod 4, divisible by 73 if *n* == 2 mod 4|http://factordb.com/index.php?query=19*27%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|29|{2}{5}|(2×29<sup>*m*+*n*</sup>+3×29<sup>*n*</sup>−5)/28|always divisible by some element of {2,3,5}<br>(nonlinear family)<br>divisible by 2 if *n* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *m* is even|–|
|29|{5}{2}|(5×29<sup>*m*+*n*</sup>−3×29<sup>*n*</sup>−2)/28|always divisible by some element of {2,3,5}<br>(nonlinear family)<br>divisible by 2 if *m* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *n* is even|–|
|29|{M}{P}|(22×29<sup>*m*+*n*</sup>+3×29<sup>*n*</sup>−25)/28|always divisible by some element of {2,3,5}<br>(nonlinear family)<br>divisible by 2 if *n* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *m* is even|–|
|29|{P}{M}|(25×29<sup>*m*+*n*</sup>−3×29<sup>*n*</sup>−22)/28|always divisible by some element of {2,3,5}<br>(nonlinear family)<br>divisible by 2 if *m* is even, divisible by 3 if *m* and *n* are both even or both odd, divisible by 5 if *n* is even|–|
|30|A{0}9J|10×30<sup>*n*+2</sup>+289 (*n* ≥ 0)|always divisible by some element of {7,13,19,31}<br>divisible by 7 if *n* == 0 mod 3, divisible by 13 if *n* == 4 mod 6, divisible by 19 if *n* == 2 mod 3, divisible by 31 if *n* is odd|http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|A{N}|(323×31<sup>*n*</sup>−23)/30 (*n* ≥ 1)|always divisible by some element of {2,13,37}<br>divisible by 2 if *n* is even, divisible by 13 if *n* == 3 mod 4, divisible by 37 if *n* == 1 mod 4|http://factordb.com/index.php?query=%28323*31%5En-23%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|{N}D|(23×31<sup>*n*+1</sup>−323)/30 (*n* ≥ 1)|always divisible by some element of {2,13,37}<br>divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 37 if *n* == 2 mod 4|http://factordb.com/index.php?query=%2823*31%5E%28n%2B1%29-323%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|D{R}|(139×31<sup>*n*</sup>−9)/10 (*n* ≥ 1)|always divisible by some element of {2,13,37}<br>divisible by 2 if *n* is odd, divisible by 13 if *n* == 0 mod 4, divisible by 37 if *n* == 2 mod 4|http://factordb.com/index.php?query=%28139*31%5En-9%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|{R}E|(9×31<sup>*n*+1</sup>−139)/10 (*n* ≥ 1)|always divisible by some element of {2,13,37}<br>divisible by 2 if *n* is even, divisible by 13 if *n* == 3 mod 4, divisible by 37 if *n* == 1 mod 4|http://factordb.com/index.php?query=%289*31%5E%28n%2B1%29-139%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|O{5}|(145×31<sup>*n*</sup>−1)/6 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}<br>divisible by 2 if *n* is even, divisible by 3 if *n* == 0 mod 3, divisible by 7 if *n* == 1 mod 6, divisible by 19 if *n* == 5 mod 6|http://factordb.com/index.php?query=%28145*31%5En-1%29%2F6&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|J{T}|(599×31<sup>*n*</sup>−29)/30 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}<br>divisible by 2 if *n* is odd, divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 2 mod 6, divisible by 19 if *n* == 0 mod 6|http://factordb.com/index.php?query=%28599*31%5En-29%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|{T}A|(29×31<sup>*n*+1</sup>−599)/30 (*n* ≥ 1)|always divisible by some element of {2,3,7,19}<br>divisible by 2 if *n* is even, divisible by 3 if *n* == 1 mod 3, divisible by 7 if *n* == 3 mod 6, divisible by 19 if *n* == 5 mod 6|http://factordb.com/index.php?query=%2829*31%5E%28n%2B1%29-599%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|8{0}V|8×32<sup>*n*+1</sup>+31 (*n* ≥ 0)|always divisible by some element of {3,5,41}<br>divisible by 3 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 41 if *n* == 0 mod 4|http://factordb.com/index.php?query=8*32%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|{G}L|(16×32<sup>*n*+1</sup>+139)/31 (*n* ≥ 1)|always divisible by some element of {3,5,41}<br>divisible by 3 if *n* is even, divisible by 5 if *n* == 3 mod 4, divisible by 41 if *n* == 1 mod 4|http://factordb.com/index.php?query=%2816*32%5E%28n%2B1%29%2B139%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{1}|(9<sup>*n*</sup>−1)/8 (*n* ≥ 2)|difference-of-two-squares factorization<br>(9<sup>*n*</sup>−1)/8 = (3<sup>*n*</sup>−1) × (3<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|1{0}1|8<sup>*n*+1</sup>+1 (*n* ≥ 0)|sum-of-two-cubes factorization<br>8<sup>*n*+1</sup>+1 = (2<sup>*n*+1</sup>+1) × (4<sup>*n*+1</sup>−2<sup>*n*+1</sup>+1)|http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|3{1}|(25×9<sup>*n*</sup>−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization<br>(25×9<sup>*n*</sup>−1)/8 = (5×3<sup>*n*</sup>−1) × (5×3<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|3{8}|4×9<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-squares factorization<br>4×9<sup>*n*</sup>−1 = (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)|http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|{8}5|9<sup>*n*+1</sup>−4 (*n* ≥ 1)|difference-of-two-squares factorization<br>9<sup>*n*+1</sup>−4 = (3<sup>*n*+1</sup>−2) × (3<sup>*n*+1</sup>+2)|http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|3{8}35|4×9<sup>*n*+2</sup>−49 (*n* ≥ 0)|difference-of-two-squares factorization<br>4×9<sup>*n*+2</sup>−49 = (2×3<sup>*n*+2</sup>−7) × (2×3<sup>*n*+2</sup>+7)|http://factordb.com/index.php?query=4*9%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|8{F}|9×16<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-squares factorization<br>9×16<sup>*n*</sup>−1 = (3×4<sup>*n*</sup>−1) × (3×4<sup>*n*</sup>+1)|http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{F}7|16<sup>*n*+1</sup>−9 (*n* ≥ 1)|difference-of-two-squares factorization<br>16<sup>*n*+1</sup>−9 = (4<sup>*n*+1</sup>−3) × (4<sup>*n*+1</sup>+3)|http://factordb.com/index.php?query=16%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{4}1|(4×16<sup>*n*+1</sup>−49)/15 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×16<sup>*n*+1</sup>−49)/15 = (2×4<sup>*n*+1</sup>−7) × (2×4<sup>*n*+1</sup>+7) / 15|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-49%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|B{4}1|(169×16<sup>*n*+1</sup>−49)/15 (*n* ≥ 0)|difference-of-two-squares factorization<br>(169×16<sup>*n*+1</sup>−49)/15 = (13×4<sup>*n*+1</sup>−7) × (13×4<sup>*n*+1</sup>+7) / 15|http://factordb.com/index.php?query=%28169*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|1{5}|(4×16<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×16<sup>*n*</sup>−1)/3 = (2×4<sup>*n*</sup>−1) × (2×4<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|8{5}|(25×16<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(25×16<sup>*n*</sup>−1)/3 = (5×4<sup>*n*</sup>−1) × (5×4<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%2825*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|10{5}|(49×16<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(49×16<sup>*n*</sup>−1)/3 = (7×4<sup>*n*</sup>−1) × (7×4<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%2849*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|A1{5}|(484×16<sup>*n*</sup>−1)/3 (*n* ≥ 0)|difference-of-two-squares factorization<br>(484×16<sup>*n*</sup>−1)/3 = (22×4<sup>*n*</sup>−1) × (22×4<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%28484*16%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|7{3}|(36×16<sup>*n*</sup>−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization<br>(36×16<sup>*n*</sup>−1)/5 = (6×4<sup>*n*</sup>−1) × (6×4<sup>*n*</sup>+1) / 5|http://factordb.com/index.php?query=%2836*16%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|3{F}AF|4×16<sup>*n*+2</sup>−81 (*n* ≥ 0)|difference-of-two-squares factorization<br>4×16<sup>*n*+2</sup>−81 = (2×4<sup>*n*+2</sup>−9) × (2×4<sup>*n*+2</sup>+9)|http://factordb.com/index.php?query=4*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|30{F}AF|49×16<sup>*n*+2</sup>−81 (*n* ≥ 0)|difference-of-two-squares factorization<br>49×16<sup>*n*+2</sup>−81 = (7×4<sup>*n*+2</sup>−9) × (7×4<sup>*n*+2</sup>+9)|http://factordb.com/index.php?query=49*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|3{F}A0F|4×16<sup>*n*+3</sup>−1521 (*n* ≥ 0)|difference-of-two-squares factorization<br>4×16<sup>*n*+3</sup>−1521 = (2×4<sup>*n*+3</sup>−39) × (2×4<sup>*n*+3</sup>+39)|http://factordb.com/index.php?query=4*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|30{F}A0F|49×16<sup>*n*+3</sup>−1521 (*n* ≥ 0)|difference-of-two-squares factorization<br>49×16<sup>*n*+3</sup>−1521 = (7×4<sup>*n*+3</sup>−39) × (7×4<sup>*n*+3</sup>+39)|http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{3}23|(16<sup>*n*+2</sup>−81)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(16<sup>*n*+2</sup>−81)/5 = (4<sup>*n*+2</sup>−9) × (4<sup>*n*+2</sup>+9) / 5<br>(in fact, difference-of-4th-powers factorization)<br>(16<sup>*n*+2</sup>−81)/5 = (2<sup>*n*+2</sup>−3) × (2<sup>*n*+2</sup>+3) × (4<sup>*n*+2</sup>+9) / 5|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{5}45|(16<sup>*n*+2</sup>−49)/3 (*n* ≥ 0)|difference-of-two-squares factorization<br>(16<sup>*n*+2</sup>−49)/3 = (4<sup>*n*+2</sup>−7) × (4<sup>*n*+2</sup>+7) / 3|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{C}B|(4×16<sup>*n*+1</sup>−9)/5 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×16<sup>*n*+1</sup>−9)/5 = (2×4<sup>*n*+1</sup>−3) × (2×4<sup>*n*+1</sup>+3) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-9%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{C}D|(4×16<sup>*n*+1</sup>+1)/5 (*n* ≥ 1)|Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup><br>(4×16<sup>*n*+1</sup>+1)/5 = (2×4<sup>*n*+1</sup>−2×2<sup>*n*+1</sup>+1) × (2×4<sup>*n*+1</sup>+2×2<sup>*n*+1</sup>+1) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{C}DD|(4×16<sup>*n*+2</sup>+81)/5 (*n* ≥ 0)|Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup><br>(4×16<sup>*n*+2</sup>+81)/5 = (2×4<sup>*n*+2</sup>−6×2<sup>*n*+2</sup>+9) × (2×4<sup>*n*+2</sup>+6×2<sup>*n*+2</sup>+9) / 5|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{1}|(25<sup>*n*</sup>−1)/24 (*n* ≥ 2)|difference-of-two-squares factorization<br>(25<sup>*n*</sup>−1)/24 = (5<sup>*n*</sup>−1) × (5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|2{1}|(49×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(49×25<sup>*n*</sup>−1)/24 = (7×5<sup>*n*</sup>−1) × (7×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|5{1}|(121×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(121×25<sup>*n*</sup>−1)/24 = (11×5<sup>*n*</sup>−1) × (11×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|7{1}|(169×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(169×25<sup>*n*</sup>−1)/24 = (13×5<sup>*n*</sup>−1) × (13×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28169*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|C{1}|(289×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(289×25<sup>*n*</sup>−1)/24 = (17×5<sup>*n*</sup>−1) × (17×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28289*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|F{1}|(361×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(361×25<sup>*n*</sup>−1)/24 = (19×5<sup>*n*</sup>−1) × (19×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28361*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|M{1}|(529×25<sup>*n*</sup>−1)/24 (*n* ≥ 1)|difference-of-two-squares factorization<br>(529×25<sup>*n*</sup>−1)/24 = (23×5<sup>*n*</sup>−1) × (23×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%28529*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|7C{1}|(4489×25<sup>*n*</sup>−1)/24 (*n* ≥ 0)|difference-of-two-squares factorization<br>(4489×25<sup>*n*</sup>−1)/24 = (67×5<sup>*n*</sup>−1) × (67×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%284489*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|D5{1}|(7921×25<sup>*n*</sup>−1)/24 (*n* ≥ 0)|difference-of-two-squares factorization<br>(7921×25<sup>*n*</sup>−1)/24 = (89×5<sup>*n*</sup>−1) × (89×5<sup>*n*</sup>+1) / 24|http://factordb.com/index.php?query=%287921*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|1{3}|(9×25<sup>*n*</sup>−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization<br>(9×25<sup>*n*</sup>−1)/8 = (3×5<sup>*n*</sup>−1) × (3×5<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%289*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|1{8}|(4×25<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×25<sup>*n*</sup>−1)/3 = (2×5<sup>*n*</sup>−1) × (2×5<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%284*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|5{8}|(16×25<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(16×25<sup>*n*</sup>−1)/3 = (4×5<sup>*n*</sup>−1) × (4×5<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%2816*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|A{3}|(81×25<sup>*n*</sup>−1)/8 (*n* ≥ 1)|difference-of-two-squares factorization<br>(81×25<sup>*n*</sup>−1)/8 = (9×5<sup>*n*</sup>−1) × (9×5<sup>*n*</sup>+1) / 8|http://factordb.com/index.php?query=%2881*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|L{8}|(64×25<sup>*n*</sup>−1)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(64×25<sup>*n*</sup>−1)/3 = (8×5<sup>*n*</sup>−1) × (8×5<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%2864*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{3}2|(25<sup>*n*+1</sup>−9)/8 (*n* ≥ 1)|difference-of-two-squares factorization<br>(25<sup>*n*+1</sup>−9)/8 = (5<sup>*n*+1</sup>−3) × (5<sup>*n*+1</sup>+3) / 8|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-9%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{8}3|(25<sup>*n*+1</sup>−16)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(25<sup>*n*+1</sup>−16)/3 = (5<sup>*n*+1</sup>−4) × (5<sup>*n*+1</sup>+4) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-16%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{8}7|(25<sup>*n*+1</sup>−4)/3 (*n* ≥ 1)|difference-of-two-squares factorization<br>(25<sup>*n*+1</sup>−4)/3 = (5<sup>*n*+1</sup>−2) × (5<sup>*n*+1</sup>+2) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{3}2I|(25<sup>*n*+2</sup>−81)/8 (*n* ≥ 0)|difference-of-two-squares factorization<br>(25<sup>*n*+2</sup>−81)/8 = (5<sup>*n*+2</sup>−9) × (5<sup>*n*+2</sup>+9) / 8|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-81%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{8}5I|(25<sup>*n*+2</sup>−196)/3 (*n* ≥ 0)|difference-of-two-squares factorization<br>(25<sup>*n*+2</sup>−196)/3 = (5<sup>*n*+2</sup>−14) × (5<sup>*n*+2</sup>+14) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-196%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{8}7C|(25<sup>*n*+2</sup>−64)/3 (*n* ≥ 0)|difference-of-two-squares factorization<br>(25<sup>*n*+2</sup>−64)/3 = (5<sup>*n*+2</sup>−8) × (5<sup>*n*+2</sup>+8) / 3|http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-64%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|8{0}1|8×27<sup>*n*+1</sup>+1 (*n* ≥ 0)|sum-of-two-cubes factorization<br>8×27<sup>*n*+1</sup>+1 = (2×3<sup>*n*+1</sup>+1) × (4×9<sup>*n*+1</sup>−2×3<sup>*n*+1</sup>+1)|http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|1{0}8|27<sup>*n*+1</sup>+8 (*n* ≥ 0)|sum-of-two-cubes factorization<br>27<sup>*n*+1</sup>+8 = (3<sup>*n*+1</sup>+2) × (9<sup>*n*+1</sup>−2×3<sup>*n*+1</sup>+4)|http://factordb.com/index.php?query=27%5E%28n%2B1%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|{D}E|(27<sup>*n*+1</sup>+1)/2 (*n* ≥ 1)|sum-of-two-cubes factorization<br>(27<sup>*n*+1</sup>+1)/2 = (3<sup>*n*+1</sup>+1) × (9<sup>*n*+1</sup>−3<sup>*n*+1</sup>+1) / 2|http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|7{Q}|8×27<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-cubes factorization<br>8×27<sup>*n*</sup>−1 = (2×3<sup>*n*</sup>−1) × (4×9<sup>*n*</sup>+2×3<sup>*n*</sup>+1)|http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|{Q}J|27<sup>*n*+1</sup>−8 (*n* ≥ 1)|difference-of-two-cubes factorization<br>27<sup>*n*+1</sup>−8 = (3<sup>*n*+1</sup>−2) × (9<sup>*n*+1</sup>+2×3<sup>*n*+1</sup>+4)|http://factordb.com/index.php?query=27%5E%28n%2B1%29-8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|9{G}|(125×27<sup>*n*</sup>−8)/13 (*n* ≥ 1)|difference-of-two-cubes factorization<br>(125×27<sup>*n*</sup>−8)/13 = (5×3<sup>*n*</sup>−2) × (25×9<sup>*n*</sup>+10×3<sup>*n*</sup>+4) / 13|http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|1{0}1|32<sup>*n*+1</sup>+1 (*n* ≥ 0)|sum-of-two-5th-powers factorization<br>32<sup>*n*+1</sup>+1 = (2<sup>*n*+1</sup>+1) × (16<sup>*n*+1</sup>−8<sup>*n*+1</sup>+4<sup>*n*+1</sup>−2<sup>*n*+1</sup>+1)|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|{1}|(32<sup>*n*</sup>−1)/31 (*n* ≥ 2)|difference-of-two-5th-powers factorization<br>(32<sup>*n*</sup>−1)/31 = (2<sup>*n*</sup>−1) × (16<sup>*n*</sup>+8<sup>*n*</sup>+4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 31|http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|3{7}|(16×36<sup>*n*</sup>−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization<br>(16×36<sup>*n*</sup>−1)/5 = (4×6<sup>*n*</sup>−1) × (4×6<sup>*n*</sup>+1) / 5|http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|3{Z}|4×36<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-squares factorization<br>4×36<sup>*n*</sup>−1 = (2×6<sup>*n*</sup>−1) × (2×6<sup>*n*</sup>+1)|http://factordb.com/index.php?query=4*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|8{Z}|9×36<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-squares factorization<br>9×36<sup>*n*</sup>−1 = (3×6<sup>*n*</sup>−1) × (3×6<sup>*n*</sup>+1)|http://factordb.com/index.php?query=9*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|O{Z}|25×36<sup>*n*</sup>−1 (*n* ≥ 1)|difference-of-two-squares factorization<br>25×36<sup>*n*</sup>−1 = (5×6<sup>*n*</sup>−1) × (5×6<sup>*n*</sup>+1)|http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{Z}B|36<sup>*n*+1</sup>−25 (*n* ≥ 1)|difference-of-two-squares factorization<br>36<sup>*n*+1</sup>−25 = (6<sup>*n*+1</sup>−5) × (6<sup>*n*+1</sup>+5)|http://factordb.com/index.php?query=36%5E%28n%2B1%29-25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|8{Z}B|9×36<sup>*n*+1</sup>−25 (*n* ≥ 0)|difference-of-two-squares factorization<br>9×36<sup>*n*+1</sup>−25 = (3×6<sup>*n*+1</sup>−5) × (3×6<sup>*n*+1</sup>+5)|http://factordb.com/index.php?query=9*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|F{Z}B|16×36<sup>*n*+1</sup>−25 (*n* ≥ 0)|difference-of-two-squares factorization<br>16×36<sup>*n*+1</sup>−25 = (4×6<sup>*n*+1</sup>−5) × (4×6<sup>*n*+1</sup>+5)|http://factordb.com/index.php?query=16*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{Z}RZ|36<sup>*n*+2</sup>−289 (*n* ≥ 0)|difference-of-two-squares factorization<br>36<sup>*n*+2</sup>−289 = (6<sup>*n*+2</sup>−17) × (6<sup>*n*+2</sup>+17)|http://factordb.com/index.php?query=36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|O{Z}RZ|25×36<sup>*n*+2</sup>−289 (*n* ≥ 0)|difference-of-two-squares factorization<br>25×36<sup>*n*+2</sup>−289 = (5×6<sup>*n*+2</sup>−17) × (5×6<sup>*n*+2</sup>+17)|http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|O{5}|(169×36<sup>*n*</sup>−1)/7 (*n* ≥ 1)|difference-of-two-squares factorization<br>(169×36<sup>*n*</sup>−1)/7 = (13×6<sup>*n*</sup>−1) × (13×6<sup>*n*</sup>+1) / 7|http://factordb.com/index.php?query=%28169*36%5En-1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|O{7}|(121×36<sup>*n*</sup>−1)/5 (*n* ≥ 1)|difference-of-two-squares factorization<br>(121×36<sup>*n*</sup>−1)/5 = (11×6<sup>*n*</sup>−1) × (11×6<sup>*n*</sup>+1) / 5|http://factordb.com/index.php?query=%28121*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{9}1|(9×36<sup>*n*+1</sup>−289)/35 (*n* ≥ 1)|difference-of-two-squares factorization<br>(9×36<sup>*n*+1</sup>−289)/35 = (3×6<sup>*n*+1</sup>−17) × (3×6<sup>*n*+1</sup>+17) / 35|http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-289%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|T{9}1|(1024×36<sup>*n*+1</sup>−289)/35 (*n* ≥ 0)|difference-of-two-squares factorization<br>(1024×36<sup>*n*+1</sup>−289)/35 = (32×6<sup>*n*+1</sup>−17) × (32×6<sup>*n*+1</sup>+17) / 35|http://factordb.com/index.php?query=%281024*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{G}D|(16×36<sup>*n*+1</sup>−121)/35 (*n* ≥ 1)|difference-of-two-squares factorization<br>(16×36<sup>*n*+1</sup>−121)/35 = (4×6<sup>*n*+1</sup>−11) × (4×6<sup>*n*+1</sup>+11) / 35|http://factordb.com/index.php?query=%2816*36%5E%28n%2B1%29-121%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{G}8D|(16×36<sup>*n*+2</sup>−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization<br>(16×36<sup>*n*+2</sup>−10201)/35 = (4×6<sup>*n*+2</sup>−101) × (4×6<sup>*n*+2</sup>+101) / 35|http://factordb.com/index.php?query=%2816*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|R{G}D|(961×36<sup>*n*+1</sup>−121)/35 (*n* ≥ 0)|difference-of-two-squares factorization<br>(961×36<sup>*n*+1</sup>−121)/35 = (31×6<sup>*n*+1</sup>−11) × (31×6<sup>*n*+1</sup>+11) / 35|http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|3{G}8D|(121×36<sup>*n*+2</sup>−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization<br>(121×36<sup>*n*+2</sup>−10201)/35 = (11×6<sup>*n*+2</sup>−101) × (11×6<sup>*n*+2</sup>+101) / 35|http://factordb.com/index.php?query=%28121*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|R{G}8D|(961×36<sup>*n*+2</sup>−10201)/35 (*n* ≥ 0)|difference-of-two-squares factorization<br>(961×36<sup>*n*+2</sup>−10201)/35 = (31×6<sup>*n*+2</sup>−101) × (31×6<sup>*n*+2</sup>+101) / 35|http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{K}H|(4×36<sup>*n*+1</sup>−25)/7 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×36<sup>*n*+1</sup>−25)/7 = (2×6<sup>*n*+1</sup>−5) × (2×6<sup>*n*+1</sup>+5) / 7|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|B{K}H|(81×36<sup>*n*+1</sup>−25)/7 (*n* ≥ 0)|difference-of-two-squares factorization<br>(81×36<sup>*n*+1</sup>−25)/7 = (9×6<sup>*n*+1</sup>−5) × (9×6<sup>*n*+1</sup>+5) / 7|http://factordb.com/index.php?query=%2881*36%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|3{K}IH|(25×36<sup>*n*+2</sup>−529)/7 (*n* ≥ 0)|difference-of-two-squares factorization<br>(25×36<sup>*n*+2</sup>−529)/7 = (5×6<sup>*n*+2</sup>−23) × (5×6<sup>*n*+2</sup>+23) / 7|http://factordb.com/index.php?query=%2825*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|B{K}IH|(81×36<sup>*n*+2</sup>−529)/7 (*n* ≥ 0)|difference-of-two-squares factorization<br>(81×36<sup>*n*+2</sup>−529)/7 = (9×6<sup>*n*+2</sup>−23) × (9×6<sup>*n*+2</sup>+23) / 7|http://factordb.com/index.php?query=%2881*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{S}J|(4×36<sup>*n*+1</sup>−49)/5 (*n* ≥ 1)|difference-of-two-squares factorization<br>(4×36<sup>*n*+1</sup>−49)/5 = (2×6<sup>*n*+1</sup>−7) × (2×6<sup>*n*+1</sup>+7) / 5|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-49%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{S}IJ|(4×36<sup>*n*+2</sup>−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(4×36<sup>*n*+2</sup>−1849)/5 = (2×6<sup>*n*+2</sup>−43) × (2×6<sup>*n*+2</sup>+43) / 5|http://factordb.com/index.php?query=%284*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|1{S}J|(9×36<sup>*n*+1</sup>−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(9×36<sup>*n*+1</sup>−49)/5 = (3×6<sup>*n*+1</sup>−7) × (3×6<sup>*n*+1</sup>+7) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|C{S}J|(64×36<sup>*n*+1</sup>−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(64×36<sup>*n*+1</sup>−49)/5 = (8×6<sup>*n*+1</sup>−7) × (8×6<sup>*n*+1</sup>+7) / 5|http://factordb.com/index.php?query=%2864*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|X{S}J|(169×36<sup>*n*+1</sup>−49)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(169×36<sup>*n*+1</sup>−49)/5 = (13×6<sup>*n*+1</sup>−7) × (13×6<sup>*n*+1</sup>+7) / 5|http://factordb.com/index.php?query=%28169*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|1{S}GJ|(9×36<sup>*n*+2</sup>−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(9×36<sup>*n*+2</sup>−2209)/5 = (3×6<sup>*n*+2</sup>−47) × (3×6<sup>*n*+2</sup>+47) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|9{S}GJ|(49×36<sup>*n*+2</sup>−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(49×36<sup>*n*+2</sup>−2209)/5 = (7×6<sup>*n*+2</sup>−47) × (7×6<sup>*n*+2</sup>+47) / 5|http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|C{S}GJ|(64×36<sup>*n*+2</sup>−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(64×36<sup>*n*+2</sup>−2209)/5 = (8×6<sup>*n*+2</sup>−47) × (8×6<sup>*n*+2</sup>+47) / 5|http://factordb.com/index.php?query=%2864*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|X{S}GJ|(169×36<sup>*n*+2</sup>−2209)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(169×36<sup>*n*+2</sup>−2209)/5 = (13×6<sup>*n*+2</sup>−47) × (13×6<sup>*n*+2</sup>+47) / 5|http://factordb.com/index.php?query=%28169*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|1{S}IJ|(9×36<sup>*n*+2</sup>−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(9×36<sup>*n*+2</sup>−1849)/5 = (3×6<sup>*n*+2</sup>−43) × (3×6<sup>*n*+2</sup>+43) / 5|http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|9{S}IJ|(49×36<sup>*n*+2</sup>−1849)/5 (*n* ≥ 0)|difference-of-two-squares factorization<br>(49×36<sup>*n*+2</sup>−1849)/5 = (7×6<sup>*n*+2</sup>−43) × (7×6<sup>*n*+2</sup>+43) / 5|http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|8{D}|9×14<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, 9×14<sup>*n*</sup>−1 = (3×14<sup>*n*/2</sup>−1) × (3×14<sup>*n*/2</sup>+1) if *n* is even|http://factordb.com/index.php?query=9*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|12|{B}9B|12<sup>*n*+2</sup>−25 (*n* ≥ 0)|combine of factor 13 and difference-of-two-squares factorization<br>divisible by 13 if *n* is odd, 12<sup>*n*+2</sup>−25 = (12<sup>(*n*+2)/2</sup>−5) × (12<sup>(*n*+2)/2</sup>+5) if *n* is even|http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|{D}5|14<sup>*n*+1</sup>−9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is even, 14<sup>*n*+1</sup>−9 = (14<sup>(*n*+1)/2</sup>−3) × (14<sup>(*n*+1)/2</sup>+3) if *n* is odd|http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|1{9}|(25×17<sup>*n*</sup>−9)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is odd, (25×17<sup>*n*</sup>−9)/16 = (5×17<sup>*n*/2</sup>−3) × (5×17<sup>*n*/2</sup>+3) / 16 if *n* is even|http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|7{9}|(121×17<sup>*n*</sup>−9)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is odd, (121×17<sup>*n*</sup>−9)/16 = (11×17<sup>*n*/2</sup>−3) × (11×17<sup>*n*/2</sup>+3) / 16 if *n* is even|http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|{9}2|(9×17<sup>*n*+1</sup>−121)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is even, (9×17<sup>*n*+1</sup>−121)/16 = (3×17<sup>(*n*+1)/2</sup>−11) × (3×17<sup>(*n*+1)/2</sup>+11) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-121%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|{9}8|(9×17<sup>*n*+1</sup>−25)/16 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is even, (9×17<sup>*n*+1</sup>−25)/16 = (3×17<sup>(*n*+1)/2</sup>−5) × (3×17<sup>(*n*+1)/2</sup>+5) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|19|1{6}|(4×19<sup>*n*</sup>−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (4×19<sup>*n*</sup>−1)/3 = (2×19<sup>*n*/2</sup>−1) × (2×19<sup>*n*/2</sup>+1) / 3 if *n* is even|http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|19|{6}5|(19<sup>*n*+1</sup>−4)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is even, (19<sup>*n*+1</sup>−4)/3 = (19<sup>(*n*+1)/2</sup>−2) × (19<sup>(*n*+1)/2</sup>+2) / 3 if *n* is odd|http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|19|7{2}|(64×19<sup>*n*</sup>−1)/9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (64×19<sup>*n*</sup>−1)/9 = (8×19<sup>*n*/2</sup>−1) × (8×19<sup>*n*/2</sup>+1) / 9 if *n* is even|http://factordb.com/index.php?query=%2864*19%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|19|89{6}|(484×19<sup>*n*</sup>−1)/3 (*n* ≥ 0)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (484×19<sup>*n*</sup>−1)/3 = (22×19<sup>*n*/2</sup>−1) × (22×19<sup>*n*/2</sup>+1) / 3 if *n* is even|http://factordb.com/index.php?query=%28484*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3{N}|4×24<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, 4×24<sup>*n*</sup>−1 = (2×24<sup>*n*/2</sup>−1) × (2×24<sup>*n*/2</sup>+1) if *n* is even|http://factordb.com/index.php?query=4*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|5{N}|6×24<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is even, 6×24<sup>*n*</sup>−1 = (12×24<sup>(*n*−1)/2</sup>−1) × (12×24<sup>(*n*−1)/2</sup>+1) if *n* is odd|http://factordb.com/index.php?query=6*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|8{N}|9×24<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, 9×24<sup>*n*</sup>−1 = (3×24<sup>*n*/2</sup>−1) × (3×24<sup>*n*/2</sup>+1) if *n* is even|http://factordb.com/index.php?query=9*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|{6}1|(6×24<sup>*n*+1</sup>−121)/23 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (6×24<sup>*n*+1</sup>−121)/23 = (12×24<sup>*n*/2</sup>−11) × (12×24<sup>*n*/2</sup>+11) / 23 if *n* is even|http://factordb.com/index.php?query=%286*24%5E%28n%2B1%29-121%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|{N}LN|24<sup>*n*+2</sup>−49 (*n* ≥ 0)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, 24<sup>*n*+2</sup>−49 = (24<sup>(*n*+2)/2</sup>−7) × (24<sup>(*n*+2)/2</sup>+7) if *n* is even|http://factordb.com/index.php?query=24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|F{W}|16×33<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization<br>divisible by 17 if *n* is odd, 16×33<sup>*n*</sup>−1 = (4×33<sup>*n*/2</sup>−1) × (4×33<sup>*n*/2</sup>+1) if *n* is even|http://factordb.com/index.php?query=16*33%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|{W}H|33<sup>*n*+1</sup>−16 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization<br>divisible by 17 if *n* is even, 33<sup>*n*+1</sup>−16 = (33<sup>(*n*+1)/2</sup>−4) × (33<sup>(*n*+1)/2</sup>+4) if *n* is odd|http://factordb.com/index.php?query=33%5E%28n%2B1%29-16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|3{P}|(121×33<sup>*n*</sup>−25)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is odd, (121×33<sup>*n*</sup>−25)/32 = (11×33<sup>*n*/2</sup>−5) × (11×33<sup>*n*/2</sup>+5) / 32 if *n* is even|http://factordb.com/index.php?query=%28121*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|D{P}|(441×33<sup>*n*</sup>−25)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is odd, (441×33<sup>*n*</sup>−25)/32 = (21×33<sup>*n*/2</sup>−5) × (21×33<sup>*n*/2</sup>+5) / 32 if *n* is even|http://factordb.com/index.php?query=%28441*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|{9}4|(9×33<sup>*n*+1</sup>−169)/32 (*n* ≥ 1)|combine of factor 2 and difference-of-two-squares factorization<br>divisible by 2 if *n* is even, (9×33<sup>*n*+1</sup>−169)/32 = (3×33<sup>(*n*+1)/2</sup>−13) × (3×33<sup>(*n*+1)/2</sup>+13) / 32 if *n* is odd|http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-169%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|33|{I}H|(9×33<sup>*n*+1</sup>−25)/16 (*n* ≥ 1)|combine of factor 17 and difference-of-two-squares factorization<br>divisible by 17 if *n* is even, (9×33<sup>*n*+1</sup>−25)/16 = (3×33<sup>(*n*+1)/2</sup>−5) × (3×33<sup>(*n*+1)/2</sup>+5) / 16 if *n* is odd|http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|1{B}|(4×34<sup>*n*</sup>−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (4×34<sup>*n*</sup>−1)/3 = (2×34<sup>*n*/2</sup>−1) × (2×34<sup>*n*/2</sup>+1) / 3 if *n* is even|http://factordb.com/index.php?query=%284*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|G{B}|(49×34<sup>*n*</sup>−1)/3 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (49×34<sup>*n*</sup>−1)/3 = (7×34<sup>*n*/2</sup>−1) × (7×34<sup>*n*/2</sup>+1) / 3 if *n* is even|http://factordb.com/index.php?query=%2849*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|1M{B}|(169×34<sup>*n*</sup>−1)/3 (*n* ≥ 0)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (169×34<sup>*n*</sup>−1)/3 = (13×34<sup>*n*/2</sup>−1) × (13×34<sup>*n*/2</sup>+1) / 3 if *n* is even|http://factordb.com/index.php?query=%28169*34%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|G{1}|(529×34<sup>*n*</sup>−1)/33 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (529×34<sup>*n*</sup>−1)/33 = (23×34<sup>*n*/2</sup>−1) × (23×34<sup>*n*/2</sup>+1) / 33 if *n* is even<br>(special example, as the numbers with length ≥ 14 in this family contain "prime > *b*" subsequence, this prime is 1<sub>13</sub>)|http://factordb.com/index.php?query=%28529*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|V{1}|(1024×34<sup>*n*</sup>−1)/33 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (1024×34<sup>*n*</sup>−1)/33 = (32×34<sup>*n*/2</sup>−1) × (32×34<sup>*n*/2</sup>+1) / 33 if *n* is even<br>(special example, as the numbers with length ≥ 14 in this family contain "prime > *b*" subsequence, this prime is 1<sub>13</sub>)|http://factordb.com/index.php?query=%281024*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|D{3}|(144×34<sup>*n*</sup>−1)/11 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, (144×34<sup>*n*</sup>−1)/11 = (12×34<sup>*n*/2</sup>−1) × (12×34<sup>*n*/2</sup>+1) / 11 if *n* is even|http://factordb.com/index.php?query=%28144*34%5En-1%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|8{X}|9×34<sup>*n*</sup>−1 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is odd, 9×34<sup>*n*</sup>−1 = (3×34<sup>*n*/2</sup>−1) × (3×34<sup>*n*/2</sup>+1) if *n* is even|http://factordb.com/index.php?query=9*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|34|{X}P|34<sup>*n*+1</sup>−9 (*n* ≥ 1)|combine of factor 5 and difference-of-two-squares factorization<br>divisible by 5 if *n* is even, 34<sup>*n*+1</sup>−9 = (34<sup>(*n*+1)/2</sup>−3) × (34<sup>(*n*+1)/2</sup>+3) if *n* is odd|http://factordb.com/index.php?query=34%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

Also families which contain only one very small prime > *b*: (this is because: for the case of covering congruence, an element in *S* is indeed in the family and > *b*, to make the factorizations be trivial; for the case of algebraic factorization (if the numbers are factored as *F* × *G* / *d*), *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial; for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as *F* × *G* / *d*), an element in *S* is indeed in the family and > *b* or/and *F* (or *G*) is equal to *d* and *G* (or *F*) is prime > *b*, to make the factorizations be trivial)

|*b*<br>(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*d*) form of this family (*n* is the number of digits in the "{}", also the lower bound of *n* to make the numbers > *b*)<br>(note: *d* divides *gcd*(*a*+*c*,*b*−1), but *d* need not be *gcd*(*a*+*c*,*b*−1), *d* = *gcd*(*a*+*c*,*b*−1) if and only if the numbers in the family are not divisible by some prime factor of *b*−1, i.e. the numbers in the family are coprime to *b*−1, in fact, *d* = *gcd*(*a*+*c*,*b*−1)/*gcd*(*a*+*c*,*b*−1,(largest trivial factor of the family)))|the only prime > *b* in this family|this prime > *b* written in decimal|why this family contains only this prime > *b*|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|---|---|
|9|2{7}5|(23×9<sup>*n*+1</sup>−23)/8 (*n* ≥ 0)|25|23|always divisible by 23<br>(in fact, also difference-of-two-squares factorization)<br>(23×9<sup>*n*+1</sup>−23)/8 = 23 × (3<sup>*n*+1</sup>−1) × (3<sup>*n*+1</sup>+1) / 8|http://factordb.com/index.php?query=%2823*9%5E%28n%2B1%29-23%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|5{7}2|(47×9<sup>*n*+1</sup>−47)/8 (*n* ≥ 0)|52|47|always divisible by 47<br>(in fact, also difference-of-two-squares factorization)<br>(47×9<sup>*n*+1</sup>−47)/8 = 47 × (3<sup>*n*+1</sup>−1) × (3<sup>*n*+1</sup>+1) / 8|http://factordb.com/index.php?query=%2847*9%5E%28n%2B1%29-47%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|3{7}4|(37×11<sup>*n*+1</sup>−37)/10 (*n* ≥ 0)|34|37|always divisible by 37|http://factordb.com/index.php?query=%2837*11%5E%28n%2B1%29-37%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|4{7}3|(47×11<sup>*n*+1</sup>−47)/10 (*n* ≥ 0)|43|47|always divisible by 47|http://factordb.com/index.php?query=%2847*11%5E%28n%2B1%29-47%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|2{0}J|2×27<sup>*n*+1</sup>+19 (*n* ≥ 0)|2J|73|always divisible by some element of {5,7,73}<br>divisible by 7 if *n* is odd, divisible by 5 if *n* == 2 mod 4, divisible by 73 if *n* == 0 mod 4|http://factordb.com/index.php?query=2*27%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|4|{1}|(4<sup>*n*</sup>−1)/3 (*n* ≥ 2)|11|5|difference-of-two-squares factorization<br>but 11 is prime, and 11 is the only prime > *b* in this family<br>(4<sup>*n*</sup>−1)/3 = (2<sup>*n*</sup>−1) × (2<sup>*n*</sup>+1) / 3|http://factordb.com/index.php?query=%284%5En-1%29%2F3&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|{1}|(8<sup>*n*</sup>−1)/7 (*n* ≥ 2)|111|73|difference-of-two-cubes factorization<br>but 111 is prime, and 111 is the only prime > *b* in this family<br>(8<sup>*n*</sup>−1)/7 = (2<sup>*n*</sup>−1) × (4<sup>*n*</sup>+2<sup>*n*</sup>+1) / 7|http://factordb.com/index.php?query=%288%5En-1%29%2F7&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{1}|(16<sup>*n*</sup>−1)/15 (*n* ≥ 2)|11|17|difference-of-two-squares factorization<br>but 11 is prime, and 11 is the only prime > *b* in this family<br>(16<sup>*n*</sup>−1)/15 = (4<sup>*n*</sup>−1) × (4<sup>*n*</sup>+1) / 15<br>(in fact, difference-of-4th-powers factorization)<br>(16<sup>*n*</sup>−1)/15 = (2<sup>*n*</sup>−1) × (2<sup>*n*</sup>+1) × (4<sup>*n*</sup>+1) / 15|http://factordb.com/index.php?query=%2816%5En-1%29%2F15&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|{1}|(27<sup>*n*</sup>−1)/26 (*n* ≥ 2)|111|757|difference-of-two-cubes factorization<br>but 111 is prime, and 111 is the only prime > *b* in this family<br>(27<sup>*n*</sup>−1)/26 = (3<sup>*n*</sup>−1) × (9<sup>*n*</sup>+3<sup>*n*</sup>+1) / 26|http://factordb.com/index.php?query=%2827%5En-1%29%2F26&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|{G}7|(8×27<sup>*n*+1</sup>−125)/13 (*n* ≥ 1)|G7|439|difference-of-two-cubes factorization<br>but G7 is prime, and G7 is the only prime > *b* in this family<br>(8×27<sup>*n*+1</sup>−125)/13 = (2×3<sup>*n*+1</sup>−5) × (4×9<sup>*n*+1</sup>+10×3<sup>*n*+1</sup>+25) / 13|http://factordb.com/index.php?query=%288*27%5E%28n%2B1%29-125%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{1}|(36<sup>*n*</sup>−1)/35 (*n* ≥ 2)|11|37|difference-of-two-squares factorization<br>but 11 is prime, and 11 is the only prime > *b* in this family<br>(36<sup>*n*</sup>−1)/35 = (6<sup>*n*</sup>−1) × (6<sup>*n*</sup>+1) / 35|http://factordb.com/index.php?query=%2836%5En-1%29%2F35&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

It is interesting to note that all numbers in families {1}, 2{1}, 5{1}, 7{1}, C{1}, F{1}, M{1}, 1F{1}, 27{1}, 7C{1}, D5{1} in base *b* = 25 are generalized pentagonal numbers (https://en.wikipedia.org/wiki/Pentagonal_number, https://mathworld.wolfram.com/PentagonalNumber.html, https://oeis.org/A001318), and all numbers in families {1}, 3{1}, 6{1}, 16{1} in base *b* = 9 and families 1{3}, A{3} in base *b* = 25 are triangular numbers (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://oeis.org/A000217), and all numbers in family {1} in base *b* = 4 and families 1{5}, 8{5}, 10{5}, A1{5} in base *b* = 16 and families 1{8}, 5{8}, L{8} in base *b* = 25 are generalized octagonal numbers (https://en.wikipedia.org/wiki/Octagonal_number, https://mathworld.wolfram.com/OctagonalNumber.html, https://oeis.org/A001082), since all generalized pentagonal numbers × 25 + 1 are also generalized pentagonal numbers, all triangular numbers × 9 + 1 and all triangular numbers × 25 + 3 are also triangular numbers, all generalized octagonal numbers × 4 + 1 and all generalized octagonal numbers × 16 + 5 and all generalized octagonal numbers × 25 + 8 are also generalized octagonal numbers, and all generalized pentagonal numbers (except 0, 1, 2, 5, 7) and all triangular numbers (except 0, 1, 3) and all generalized octagonal numbers (except 0, 1, 5) are composite, thus these families contain no primes > *b* (except the family {1} in base 4, which contains a prime 11 (5 in decimal) > *b*).

Some *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families could not be proven to contain no primes > *b* (by covering congruence, algebraic factorization, or combine of them) but no primes > *b* could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to test the primality of larger and larger numbers of such form and hope a prime is eventually discovered.

Many *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families contain no small primes > *b* even though they do contain very large primes. e.g. the smallest prime in base 13 family 9{5} is 95<sub>197420</sub> (http://www.primenumbers.net/prptop/searchform.php?form=%28113*13%5E197420-5%29%2F12&action=Search, http://factordb.com/index.php?id=1100000003943359311, for this prime written in base *b* = 13 see http://factordb.com/index.php?showid=1100000003943359311&base=13, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is (113×13<sup>197420</sup>−5)/12, when written in decimal it contains 219916 digits; and the smallest prime in base 16 family {3}AF is 3<sub>116137</sub>AF (http://www.primenumbers.net/prptop/searchform.php?form=%2816%5E116139%2B619%29%2F5&action=Search, http://factordb.com/index.php?id=1100000003851731988, for this prime written in base *b* = 16 see http://factordb.com/index.php?showid=1100000003851731988&base=16, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is (16<sup>116139</sup>+619)/5, when written in decimal it contains 139845 digits; and the smallest prime in base 23 family 9{E} is 9E<sub>800873</sub> (http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search, http://factordb.com/index.php?id=1100000000782858648, for this prime written in base *b* = 23 see http://factordb.com/index.php?showid=1100000000782858648&base=23, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%28106*23%5En-7%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is (106×23<sup>800873</sup>−7)/11, when written in decimal it contains 1090573 digits; and the smallest prime in base 25 family 71JD{0}1 is 71JD0<sub>458549</sub>1 (https://t5k.org/primes/page.php?id=111834, http://factordb.com/index.php?id=1100000002341496334, for this prime written in base *b* = 25 see http://factordb.com/index.php?showid=1100000002341496334&base=25, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=110488*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is 110488×25<sup>458550</sup>+1, when written in decimal it contains 641031 digits (this number can be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored); and the smallest prime in base 32 family NU{0}1 is NU0<sub>661863</sub>1 (https://t5k.org/primes/page.php?id=134216, http://factordb.com/index.php?id=1100000003813355148, for this prime written in base *b* = 32 see http://factordb.com/index.php?showid=1100000003813355148&base=32, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=766*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is 766×32<sup>661864</sup>+1, when written in decimal it contains 996208 digits (this number can be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored); and the smallest prime in base 36 family {P}SZ is P<sub>81993</sub>SZ (http://www.primenumbers.net/prptop/searchform.php?form=%285*36%5E81995%2B821%29%2F7&action=Search, http://factordb.com/index.php?id=1100000002394962083, for this prime written in base *b* = 36 see http://factordb.com/index.php?showid=1100000002394962083&base=36, and for the factorization of the numbers in the family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*) see http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show), its algebraic form is (5×36<sup>81995</sup>+821)/7, when written in decimal it contains 127609 digits. (technically, probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality tests (https://t5k.org/prove/prove2.html) were used to show these for the numbers which cannot be proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) or *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), i.e. for the ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (which have a very small chance of making an error (https://t5k.org/notes/prp_prob.html, https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf))) because all known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) run far too slowly (the CPU time (https://en.wikipedia.org/wiki/CPU_time) is longer than the life expectancy of human (https://en.wikipedia.org/wiki/Life_expectancy) for numbers > 10<sup>100000</sup>, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10<sup>500000</sup>, and longer than one quettasecond (https://en.wikipedia.org/wiki/Quetta-) for numbers > 10<sup>3000000</sup>, even if we can do 10<sup>9</sup> bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second) to run on these numbers, see https://mersenneforum.org/showpost.php?p=627117&postcount=1) to run on numbers of these sizes unless either *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) or *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) (or both) (unfortunely, none of Wikipedia, Prime Wiki, Mathworld has article for *N*−1 primality test or *N*+1 primality test, but a similar article for Pocklington primality test: https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, also see the article for the cyclotomy primality test: https://t5k.org/glossary/xpage/Cyclotomy.html) can be ≥ 1/3 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) or can be ≥ 1/4 factored and the number is not very large (say not > 10<sup>100000</sup>), or *N*<sup>*n*</sup>−1 can be ≥ 1/3 factored for a small *n*. If either *N*−1 or *N*+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington *N*−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (the *N*−1 case) or the Morrison *N*+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) (the *N*+1 case); if either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use *CHG* (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://mersenneforum.org/showpost.php?p=528149&postcount=3 and https://mersenneforum.org/showpost.php?p=603181&postcount=438), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers *N* (thus, there are no unproven probable primes *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored), *CHG* for the numbers *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large *N* (say > 10<sup>100000</sup>), for the examples of the numbers which are proven prime by *CHG*, see https://t5k.org/primes/page.php?id=126454, https://t5k.org/primes/page.php?id=131964, https://t5k.org/primes/page.php?id=123456, https://t5k.org/primes/page.php?id=130933, https://stdkmd.net/nrr/cert/1/ (search for "CHG"), https://stdkmd.net/nrr/cert/2/ (search for "CHG"), https://stdkmd.net/nrr/cert/3/ (search for "CHG"), https://stdkmd.net/nrr/cert/4/ (search for "CHG"), https://stdkmd.net/nrr/cert/5/ (search for "CHG"), https://stdkmd.net/nrr/cert/6/ (search for "CHG"), https://stdkmd.net/nrr/cert/7/ (search for "CHG"), https://stdkmd.net/nrr/cert/8/ (search for "CHG"), https://stdkmd.net/nrr/cert/9/ (search for "CHG"), http://xenon.stanford.edu/~tjw/pp/index.html (search for "CHG"), however, *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165; if neither *N*−1 nor *N*+1 can be ≥ 1/4 factored but *N*<sup>*n*</sup>−1 can be ≥ 1/3 factored for a small *n*, then we can use the cyclotomy primality test (https://t5k.org/glossary/xpage/Cyclotomy.html, https://t5k.org/prove/prove3_3.html, http://factordb.com/nmoverview.php?method=3)), i.e. it is too hard to prove primes for general numbers (https://t5k.org/glossary/xpage/OrdinaryPrime.html) of this size, but they are expected to be primes, since they are > 10<sup>25000</sup> and the probability that they are in fact composite is < 10<sup>−2000</sup>, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf).

The numbers in *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families are of the form (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) for some fixed *a*, *b*, *c* such that *a* ≥ 1, *b* ≥ 2 (*b* is the base), *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1. Except in the special case *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1 (since this special case *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1 is the only case which *N*−1 or/and *N*+1 is smooth (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683), i.e. the case *c* = 1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableProthGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php) base *b*: *a*×*b*<sup>*n*</sup>+1, they are related to generalized Sierpinski conjecture base *b* (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian))) can be easily proven prime using Pocklington *N*−1 method (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), and the case *c* = −1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Riesel prime (https://www.rieselprime.de/ziki/Riesel_prime, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php) base *b*: *a*×*b*<sup>*n*</sup>−1, they are related to generalized Riesel conjecture base *b* (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt)) can be easily proven prime using Morrison *N*+1 method (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), these primes can be proven prime using Yves Gallot's *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see https://mersenneforum.org/showpost.php?p=611607&postcount=10 and https://mersenneforum.org/showpost.php?p=235113&postcount=8 and https://mersenneforum.org/showpost.php?p=541285&postcount=4 and https://mersenneforum.org/showpost.php?p=586913&postcount=429 and https://mersenneforum.org/showpost.php?p=605958&postcount=441, also see https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (the bottom of the page, about the factorization of (71<sup>16384</sup>+1)/2−1 and (71<sup>16384</sup>+1)/2+1) and https://web.archive.org/web/20170515153924/http://bitc.bme.emory.edu/~lzhou/blogs/?p=263, also see the *README* file for *LLR* (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html), also see the (generalized) Proth/Riesel prime search page (https://t5k.org/primes/search_proth.php), i.e. there are no unproven probable primes *N* such that *N*−1 or/and *N*+1 is ≥ 1/3 factored (the Fermat number (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, http://www.fermatsearch.org/, https://64ordle.au/fermat/) *F*<sub>33</sub> = 2<sup>2<sup>33</sup></sup>+1 and the double Mersenne number (https://en.wikipedia.org/wiki/Double_Mersenne_number, https://www.rieselprime.de/ziki/Double_Mersenne_number, https://mathworld.wolfram.com/DoubleMersenneNumber.html, http://www.doublemersennes.org/) *M*<sub>*M*<sub>61</sub></sub> = 2<sup>2<sup>61</sup>−1</sup>−1 are not "unproven probable primes" (http://factordb.com/listtype.php?t=1), they are "numbers with unknown status" (http://factordb.com/listtype.php?t=2), they are too large to be primality tested or probable-primality tested, and we can only do trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) to disprove their primality, *F*<sub>33</sub> = 2<sup>2<sup>33</sup></sup>+1 is trial factored to (4.5×10<sup>17</sup>)×2<sup>35</sup>+1 (see http://www.fermatsearch.org/stat/n.php) and *M*<sub>*M*<sub>61</sub></sub> = 2<sup>2<sup>61</sup>−1</sup>−1 is trial factored to (2.7×10<sup>17</sup>)×(2<sup>61</sup>−1)+1 (see http://www.doublemersennes.org/mm61.php)), also you can compare the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the top probable primes page (http://www.primenumbers.net/prptop/prptop.php), also see https://stdkmd.net/nrr/prime/primesize.txt and https://stdkmd.net/nrr/prime/primesize.zip (see which numbers are proven primes and which numbers are only probable primes), also see https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers")), when *n* is large the known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for such a number are too inefficient to run. In this case one must resort to a probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality test (https://t5k.org/prove/prove2.html) such as a Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) or a Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html), unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html), it is possible to use a sieving process (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) to find divisors rather than using trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172), the sieving process should remove the *n* such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) has small prime factors (say < 10<sup>9</sup>) (i.e. is not 10<sup>9</sup>-rough (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)) or/and has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>), like https://stdkmd.net/nrr/1/10003.htm#prime_period and https://stdkmd.net/nrr/3/30001.htm#prime_period and https://stdkmd.net/nrr/1/13333.htm#prime_period and https://stdkmd.net/nrr/3/33331.htm#prime_period and https://stdkmd.net/nrr/1/11113.htm#prime_period and https://stdkmd.net/nrr/3/31111.htm#prime_period (we should remove the *n* with these forms, e.g. 6×*k*+1, 6×*k*+4, 15×*k*+14, 16×*k*+3, 18×*k*+14, 21×*k*+19, 22×*k*+9, 28×*k*+13, 33×*k*+21, 34×*k*+2, ... for the family 1{0}3 in decimal (since the *n* with these forms will make the number either has small prime factors or has algebraic factors (or both)), and 20.74% of the *n* will remain in the sieve file of the family 1{0}3 in decimal (20.74% is the Nash weight (or difficulty) of the family 1{0}3 in decimal), if (and only if) the family can be proven to contain no primes > *b* (or only contain finitely many primes > *b*) (by covering congruence, algebraic factorization, or combine of them), then the sieve file will be empty, and the Nash weight (or difficulty) of such family is 0%, like https://stdkmd.net/nrr/9/91113.htm#prime_period and https://stdkmd.net/nrr/9/94449.htm#prime_period and https://stdkmd.net/nrr/9/95559.htm#prime_period).

To do this, we made use of Geoffrey Reynolds' *SRSIEVE* software (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe). This program uses the baby-step giant-step (https://en.wikipedia.org/wiki/Baby-step_giant-step) algorithm (https://en.wikipedia.org/wiki/Algorithm, https://www.rieselprime.de/ziki/Algorithm, https://mathworld.wolfram.com/Algorithm.html) to find all primes *p* which divide *a*×*b*<sup>*n*</sup>+*c* where *p* and *n* lie in a specified range (https://en.wikipedia.org/wiki/Interval_(mathematics), https://mathworld.wolfram.com/Interval.html), by using discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html) to solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*<sup>*n*</sup>+*c* == 0 mod *p* (i.e. solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) *a*×*b*<sup>*n*</sup>+*c* = 0 in the finite field (https://en.wikipedia.org/wiki/Finite_field, https://mathworld.wolfram.com/FiniteField.html) *Z*<sub>*p*</sub>) (also, this program was updated so that it also removes the *n* such that *a*×*b*<sup>*n*</sup>+*c* has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-*n*th-powers factorization with odd *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>), see https://mersenneforum.org/showpost.php?p=452132&postcount=66 and https://mersenneforum.org/showpost.php?p=451337&postcount=32 and https://mersenneforum.org/showpost.php?p=232904&postcount=604 and https://mersenneforum.org/showthread.php?t=21916 and https://mersenneforum.org/showpost.php?p=383690&postcount=1 and https://mersenneforum.org/showpost.php?p=207886&postcount=253 and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c (note: for the sequence (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1), the case of "Mersenne number" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = −1 and the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) ≥ 3, and the case of "GFN" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) is either 1 or 2)). Since this program cannot handle the general case (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1 we only used it to sieve the sequence *a*×*b*<sup>*n*</sup>+*c* for primes *p* not dividing *gcd*(*a*+*c*,*b*−1), and initialized the list of candidates to not include *n* for which there is some prime *p* dividing *gcd*(*a*+*c*,*b*−1) for which *p* dividing (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when *a*, *b*, and *c* were all odd (since then 2 divides *a*×*b*<sup>*n*</sup>+*c*, but 2 may not divide (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960).

**Edit: Now the *SRSIEVE* in *MTSIEVE* (https://sourceforge.net/projects/mtsieve/, http://mersenneforum.org/rogue/mtsieve.html, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8) can handle the general case (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1, see https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/CHANGES.txt, thus now we can sieve the sequence (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) directly.**

When sieving the sequence (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) (*a* ≥ 1, *b* ≥ 2, *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1), the sieve program should: (below, *r* is a linear function of *n*, *m* is a constant like *a*, *b*, *c* (*m* ≥ 2))

1. General:
1.1. If (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*<sup>*r*</sup>−1)/(*m*−1); display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all composite *r*, and only sieve with the primes *p* == 1 mod *r*).
1.2. If (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*<sup>*r*</sup>+1)/(*m*+1); display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored algebraically or sieved with another program (remove all composite *r*, and only sieve with the primes *p* == 1 mod 2×*r*).
1.3. If (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as *m*<sup>*r*</sup>+1; display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all non-power-of-2 *r*, and no need to sieve, and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) with the primes *p* == 1 mod 2×*r*).
1.4. If (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be written as (*m*<sup>*r*</sup>+1)/2; display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all non-power-of-2 *r*, and no need to sieve, and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) with the primes *p* == 1 mod 2×*r*).
2. Remove all *n* cases:
2.1. If *a*, *b*, −*c* are all squares; remove all *n*.
2.2. If *a*, *b*, *c* are all *r*-th powers for an odd *r* > 1; remove all *n*.
2.3. If *b* and 4×*a*×*c* are both 4th powers; remove all *n*. These are Aurifeuillean factors.
3. Remove partial *n* cases:
3.1. If *a* and −*c* are both squares; remove all *n* == 0 mod 2.
3.2. If *a* and *c* are both *r*-th powers for an odd r > 1; for each such *r*, remove all *n* == 0 mod *r*.
3.3. If 4×*a*×*c* is a 4th power; remove all *n* == 0 mod 4.
3.4. If 4×*a*×*c* is a 4th power and *b* is square; remove all *n* == 0 mod 2.
3.5. If *a*×*c* and 4×*b* are both 4th powers; remove all *n* == 1 mod 2.
3.6. If *a*×*c* is a 4th power and 2×*b* is a square; remove all *n* == 2 mod 4.

#1 and #2 should all be checked first before preceding, #3.3 and #3.4 and #3.5 are more Aurifeuillean factors.

(these are exactly the *n* such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization, and (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) if and only if there is a prime *p* such that there is no *n* satisfies these two conditions simultaneously: "(*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization" and "(*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number" (if and only if (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by covering congruence, then there is a prime *p* such that there is no *n* such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number, if and only if (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by algebraic factorization, then (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) has algebraic factorization for all *n*, and if and only if (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes by combine of covering congruence and algebraic factorization, then there is a prime *p* such that there is no *n* satisfies these two conditions simultaneously: "(*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) does not have algebraic factorization" and "(*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is a *p*-rough number"), thus, if and only if (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then the sieve file of (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) will be empty) 

For examples:

|*b*<br>(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the family|the sieve program should|reason|this family corresponding to|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|---|---|
|35|{1}|(35<sup>*n*</sup>−1)/34|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that *n* is composite, and only sieve with the primes *p* == 1 mod *n*)|this form can be written as (35<sup>*n*</sup>−1)/34|1<sub>313</sub> (the ?th minimal prime in base 35)|http://factordb.com/index.php?query=%2835%5En-1%29%2F34&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|4{D}|(9×27<sup>*n*</sup>−1)/2|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that 3×*n*+2 is composite, and only sieve with the primes *p* == 1 mod 3×*n*+2)|this form can be written as (3<sup>3×*n*+2</sup>−1)/2|4D<sub>23</sub> (the 99696th minimal prime in base 27)|http://factordb.com/index.php?query=%289*27%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|1{V}|2×32<sup>*n*</sup>−1|display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all *n* such that 5×*n*+1 is composite, and only sieve with the primes *p* == 1 mod 5×*n*+1)|this form can be written as 2<sup>5×*n*+1</sup>−1|1V<sub>6</sub> (the 72855th minimal prime in base 32)|http://factordb.com/index.php?query=2*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{U}V|(6×36<sup>*n*+1</sup>)/7|display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored algebraically or sieved with another program (remove all *n* such that 2×*n*+3 is composite, and only sieve with the primes *p* == 1 mod 2×*n*+3)|this form can be written as (6<sup>2×*n*+3</sup>+1)/7|U<sub>4</sub>V (the 12765th minimal prime in base 36)|http://factordb.com/index.php?query=%286*36%5E%28n%2B1%29%2B1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|4{0}1|4×32<sup>*n*+1</sup>+1|display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that 5×*n*+7 is not power of 2, and no need to sieve, and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) with the primes *p* == 1 mod 10×*n*+14)|this form can be written as 2<sup>5×*n*+7</sup>+1|unsolved family|http://factordb.com/index.php?query=4*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|31|{F}G|(31<sup>*n*+1</sup>+1)/2|display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that *n*+1 is not power of 2, and no need to sieve, and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) with the primes *p* == 1 mod 2×*n*+2)|this form can be written as (31<sup>*n*+1</sup>+1)/2|unsolved family|http://factordb.com/index.php?query=%2831%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|4{D}E|(9×27<sup>*n*+1</sup>+1)/2|display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all *n* such that 3×*n*+5 is not power of 2, and no need to sieve, and instead use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) with the primes *p* == 1 mod 6×*n*+10)|this form can be written as (3<sup>3×*n*+5</sup>+1)/2|4D<sub>10</sub>E (the 88466th minimal prime in base 27)|http://factordb.com/index.php?query=%289*27%5E%28n%2B1%29%2B1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|3{8}|4×9<sup>*n*</sup>−1|remove all *n*|4, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|3{1}|(25×9<sup>*n*</sup>−1)/8|remove all *n*|25, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|8{F}|9×16<sup>*n*</sup>−1|remove all *n*|9, 16, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|8{0}1|8×27<sup>*n*+1</sup>+1|remove all *n*|8, 27, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|9{G}|(125×27<sup>*n*</sup>−8)/13|remove all *n*|125, 27, −8 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{C}DD|(4×16<sup>*n*+2</sup>+81)/5|remove all *n*|16 and 4×4×81 are both 4th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|{4}1|(4×8<sup>*n*+1</sup>−25)/7|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|4 and 25 are both squares|4<sub>8</sub>1 (the 70th minimal prime in base 8)|http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|{7}2L|(22<sup>*n*+2</sup>−289)/3|remove all *n* such that *n*+2 == 0 mod 2 (i.e. remove all *n* == 0 mod 2)|1 and 289 are both squares|7<sub>3815</sub>2L (the 8002nd minimal prime in base 22)|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|2E{B}|(121×23<sup>*n*</sup>−1)/2|remove all *n* == 0 mod 2|121 and 1 are both squares|2EB<sub>29583</sub> (the 65156th minimal prime in base 23)|http://factordb.com/index.php?query=%28121*23%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|{G}99|(16×20<sup>*n*+2</sup>−2809)/19|remove all *n* such that *n*+2 == 0 mod 2 (i.e. remove all *n* == 0 mod 2)|16 and 2809 are both squares|G<sub>447</sub>99 (the 3307th minimal prime in base 20)|http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|35|{Y}V|35<sup>*n*+1</sup>−4|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|1 and 4 are both squares|Y<sub>12</sub>V (the ?th minimal prime in base 35)|http://factordb.com/index.php?query=35%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|5|1{0}13|5<sup>*n*+2</sup>+8|remove all *n* such that *n*+2 == 0 mod 3 (i.e. remove all *n* == 1 mod 3)|1 and 8 are both cubes|10<sub>93</sub>13 (the 22nd minimal prime in base 5)|http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|8{0}1|8×23<sup>*n*+1</sup>+1|remove all *n* such that *n*+1 == 0 mod 3 (i.e. remove all *n* == 2 mod 3)|8 and 1 are both cubes|unsolved family|http://factordb.com/index.php?query=8*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|1{0}1F|17<sup>*n*+2</sup>+32|remove all *n* such that *n*+2 == 0 mod 5 (i.e. remove all *n* == 3 mod 5)|1 and 32 are both 5th powers|10<sub>9019</sub>1F (the 10400th minimal prime in base 17)|http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|17|79{0}1|128×17<sup>*n*+1</sup>+1|remove all *n* such that *n*+1 == 0 mod 7 (i.e. remove all *n* == 6 mod 7)|128 and 1 are both 7th powers|790<sub>224</sub>1 (the 10307th minimal prime in base 17)|http://factordb.com/index.php?query=128*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|23|4{0}1|4×23<sup>*n*+1</sup>+1|remove all *n* such that *n*+1 == 0 mod 4 (i.e. remove all *n* == 3 mod 4)|4×4×1 is a 4th power|40<sub>341</sub>1 (the 64770th minimal prime in base 23)|http://factordb.com/index.php?query=4*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|{S}T|(4×36<sup>*n*+1</sup>+1)/5|remove all *n* such that *n*+1 == 0 mod 2 (i.e. remove all *n* == 1 mod 2)|4×4×1 is a 4th power and 36 is a square|S<sub>44</sub>T (the 35018th minimal prime in base 36)|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

Also the forms satisfying both "General" and "Remove all *n* cases", in this case the program should not display a warning message on the screen and instead stop immediately (just like the forms only satisfying "Remove all *n* cases"):

|*b*<br>(2 ≤ *b* ≤ 36)|family|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the family|the sieve program should|reason|this family corresponding to|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|---|---|
|9|{1}|(9<sup>*n*</sup>−1)/8|remove all *n*|1, 9, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|1{5}|(4×16<sup>*n*</sup>−1)/3|remove all *n*|4, 16, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|25|{1}|(25<sup>*n*</sup>−1)/24|remove all *n*|1, 25, 1 are all squares|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|{1}|(32<sup>*n*</sup>−1)/31|remove all *n*|1, 32, 1 are all 5th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|{C}D|(4×16<sup>*n*</sup>+1)/5|remove all *n*|16 and 4×4×1 are both 4th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|8|1{0}1|8<sup>*n*+1</sup>+1|remove all *n*|1, 8, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|27|{D}E|(27<sup>*n*+1</sup>+1)/2|remove all *n*|1, 27, 1 are all cubes|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|32|1{0}1|32<sup>*n*+1</sup>+1|remove all *n*|1, 32, 1 are all 5th powers|ruled out as only contain composites (only count the numbers > *b*)|http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

Coordination with existing code:

* If all *n* are removed by algebraic factors for sequence (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1), program should stop immediately. This means that this family can be proven to only contain composites.
* If some *n* are removed by algebraic factors for sequence (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1), program continues sieving for removing the numbers with small prime factors.
* Program should be able to handle input of one or multiple sequences (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) at the screen or in a file. Some sequences (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) could have algebraic factors while others do not.
* Program should be able to handle an already sieved file as input, check the file for algebraic factors, remove them, and then continue sieving more deeply. Once again some sequences (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) could have algebraic factors while others do not.

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software *LLR* by Jean Penné (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3). Although undocumented, it is possible to run these two programs on numbers of the form (*a*×*b*<sup>*n*</sup>+*c*)/*d* when *d* > 1 (however, of course, the numbers (*a*×*b*<sup>*n*</sup>+*c*)/*d* with |*c*| ≠ 1 or/and *d* ≠ 1 or/and *a* > *b*<sup>*n*</sup> can only be probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1) tested; the numbers *a*×2<sup>*n*</sup>±1 (with *a* < 2<sup>*n*</sup>) are the fastest to test, *a*×2<sup>*n*</sup>+1 numbers are tested using the Proth algorithm (https://en.wikipedia.org/wiki/Proth%27s_theorem, https://www.rieselprime.de/ziki/Proth%27s_theorem, https://mathworld.wolfram.com/ProthsTheorem.html, http://www.numericana.com/answer/primes.htm#proth), *a*×2<sup>*n*</sup>−1 numbers are tested using the Lucas-Lehmer-Riesel algorithm (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test); the numbers *a*×*b*<sup>*n*</sup>±1 (with *b* > 2, *a* < *b*<sup>*n*</sup>) can also be definitely prime (https://en.wikipedia.org/wiki/Provable_prime, https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4) tested, *a*×*b*<sup>*n*</sup>+1 numbers are tested using the *N*−1 Pocklington algorithm (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), *a*×*b*<sup>*n*</sup>−1 numbers are tested using the *N*+1 Morrison algorithm (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)), so this program required no modifications. A script was also written which allowed one to run srsieve while *LLR* or *PFGW* was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by *LLR* or *PFGW* it would be removed from the list of candidates.

For the primes < 10<sup>25000</sup> for the solved or near-solved bases (bases *b* with ≤ 6 unsolved families, i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36), we employed *PRIMO* by Marcel Martin (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or *CM* by Andreas Enge (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm), two different elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementations, to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php) for the candidates for minimal prime base *b* which are > 10<sup>299</sup> and neither *N*−1 nor *N*+1 can be ≥ 1/3 factored (need *CHG* proof if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, but *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165).

**(sorry, I do not give the sieve files of this problem in the pages, for the examples of the sieve files, see http://www.noprimeleftbehind.net/crus/sieve-sierp-base3-13G-20G-50K-100K.zip (Sierpinski problem base 3, *k* = 13000000000 to 20000000000, *n* = 50000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base3-20G-30G-50K-100K.zip (Sierpinski problem base 3, *k* = 20000000000 to 30000000000, *n* = 50000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base3-30G-40G-50K-100K.zip (Sierpinski problem base 3, *k* = 30000000000 to 40000000000, *n* = 50000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base10-3M-5M.txt (Sierpinski problem base 10, *n* = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base16-1M-2M.zip (Sierpinski problem base 16, *n* = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base22-2M-5M.txt (Sierpinski problem base 22, *n* = 2000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base25-350K-1M.zip (Sierpinski problem base 25, *n* = 350000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base26-1M-3M.txt (Sierpinski problem base 26, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base27-2M-5M.txt (Sierpinski problem base 27, *n* = 2000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base28-1M-3M.txt (Sierpinski problem base 28, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base37-1M-3M.txt (Sierpinski problem base 37, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base43-1M-3M.txt (Sierpinski problem base 43, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base46-700K-1M.txt (Sierpinski problem base 46, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base48-700K-1M.txt (Sierpinski problem base 48, *n* = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base51-30K-100K.zip (Sierpinski problem base 51, *n* = 30000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base52-500K-1M.zip (Sierpinski problem base 52, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base53-700K-1M.txt (Sierpinski problem base 53, *n* = 700000 to 1000000, not include *k* = 4) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base55-1M-3M.zip (Sierpinski problem base 55, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base60-500K-1M.zip (Sierpinski problem base 60, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base61-500K-1M.txt (Sierpinski problem base 61, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-R2-2nd-conj-7M-10M.zip (2nd Riesel problem base 2, *n* = 7000000 to 10000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base3-11G-20G-100K-200K.zip (Riesel problem base 3, *k* = 11000000000 to 20000000000, *n* = 100000 to 200000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base6-5.6M-15M.txt (Riesel problem base 6, *n* = 5600000 to 15000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-300M-400M-25K-100K.zip (Riesel problem base 7, *k* = 300000000 to 400000000, *n* = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-400M-500M-25K-100K.zip (Riesel problem base 7, *k* = 400000000 to 500000000, *n* = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base10-3M-5M.txt (Riesel problem base 10, *n* = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base16-1M-2M.txt (Riesel problem base 16, *n* = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base23-2M-5M.txt (Riesel problem base 23, *n* = 2000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base28-1M-3M.txt (Riesel problem base 28, *n* = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base30-500K-1M.txt (Riesel problem base 30, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base36-400K-2M.zip (Riesel problem base 36, *n* = 400000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base37-500K-1M.txt (Riesel problem base 37, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base42-300K-500K.txt (Riesel problem base 42, *n* = 300000 to 500000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base45-500K-1M.txt (Riesel problem base 45, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base46-500K-1M.txt (Riesel problem base 46, *n* = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base55-500K-1M.txt (Riesel problem base 55, *n* = 500000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.17.txt (original minimal prime problem base 17, *n* = 1000000 to 2000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.19.txt (original minimal prime problem base 19, *n* = 707348 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.21.txt (original minimal prime problem base 21, *n* = 506720 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.25.txt (original minimal prime problem base 25, *n* = 300000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.26.txt (original minimal prime problem base 26, *n* = 486721 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.27.txt (original minimal prime problem base 27, *n* = 360000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.28.txt (original minimal prime problem base 28 (in fact also this new minimal prime problem base 28), *n* = 543202 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.29.txt (original minimal prime problem base 29, *n* = 240000 to 1000000) and https://kurtbeschorner.de/db-details-3-1M.htm (family {1} in decimal) and https://www.alfredreichlg.de/10w7/prp/ProofFile.200001-1000000.txt (family 1{0}7 in decimal) and https://www.alfredreichlg.de/10w7/prp/ProofFile.1000001-1075000.txt (family 1{0}7 in decimal) and https://oeis.org/A076336/a076336d.html (4847×2<sup>*n*</sup>+1) and http://web.archive.org/web/20050929031631/http://robin.mathi.com/28433/ (28433×2<sup>*n*</sup>+1))**

We have completely solved this problem for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 (i.e. we have found all minimal primes in these bases and proved that they are all such primes and proved that they are definitely primes (https://en.wikipedia.org/wiki/Provable_prime, http://factordb.com/listtype.php?t=4) (i.e. not merely probable primes)) (thus, currently we can complete the classification of the minimal primes in these bases, and the "minimal prime problem" for these bases are theorems (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/), they can be called "minimal prime theorems", only bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 currently have "minimal prime theorems", and like the four color theorem (https://en.wikipedia.org/wiki/Four_color_theorem, https://mathworld.wolfram.com/Four-ColorTheorem.html) and the theorem that the Ramsey number (https://en.wikipedia.org/wiki/Ramsey%27s_theorem, https://mathworld.wolfram.com/RamseyNumber.html, https://oeis.org/A212954) *R*(4,5) = 25, the harder parts of the proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/) are completed by computers instead of humans, such as the proof for base *b* = 24 (the largest base *b* which is currently completely solved, including the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set), computing data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ *b* ≤ 26) and computing the primality certificate (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php) for the largest minimal prime in base *b* = 24 (i.e. N00N<sub>8129</sub>, its algebraic form is 13249×24<sup>8131</sup>−49) (see http://factordb.com/cert.php?id=1100000003593391606 and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/certificate24_3409 for its primality certificate) are completed by computers instead of humans (I am very glad that the problem in base *b* = 24 can be completely solved, since the number 24 is an important number in number theory, see https://sites.google.com/view/24-important-number-theory and https://oeis.org/A018253 and https://math.ucr.edu/home/baez/numbers/24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_176.pdf) and https://arxiv.org/pdf/1104.5052.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_177.pdf)), the fully proof for base *b* = 24 is almost impossible to be written by hand (only bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 is possible to write the fully proof by hand), also we have completely solved this problem for bases *b* = 11, 16, 22, 30 if we allow probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1) > 10<sup>25000</sup> in place of proven primes, besides, we have completely solved this problem for bases *b* = 13, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36 (if we allow strong probable primes in place of proven primes) except the families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) (thus, currently the "minimal prime problem" for these bases are still unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/)).

The final goal of this project is finding all minimal primes and proving that these are all such primes (including the primality proving for the probable primes) in all bases 2 ≤ *b* ≤ 36, i.e. solving all families in all bases 2 ≤ *b* ≤ 36. Solving all (left) families in all bases 2 ≤ *b* ≤ 36 (and proving the primality of all probable primes in the sets of all bases 2 ≤ *b* ≤ 36) is not possible but we aim to solve many of them (and proving the primality of many of them), at least find a *probable* prime for many of them (since the smallest prime in a family may be too large (> 10<sup>25000</sup>) to be proved primality, unless its *N*−1 or/and *N*+1 can be ≥ 25% factored).

These sets of minimal primes are computed (https://en.wikipedia.org/wiki/Computing) by: make data up to linear families (i.e. only linear families left) (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc, https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpp.cc) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem) → search the left linear families to length 1000 (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpm.cc) (just like the new base script for Conjectures ‘R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://t5k.org/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/4e524f26e39cc3df98f017e8106720ba4588e981 and https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9 and https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem) → use a program like *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) the left linear families with primes *p* < 10<sup>9</sup> (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchLLR.cc) → use *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to primality test (or probable-primality test) the numbers in the sieve files (*LLR* will do the Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) with first 50 prime bases, the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html), and the strong Frobenius primality test (https://en.wikipedia.org/wiki/Strong_Frobenius_pseudoprime, https://t5k.org/glossary/xpage/FrobeniusPseudoprime.html, https://mathworld.wolfram.com/StrongFrobeniusPseudoprime.html), also for *a*×*b*<sup>*n*</sup>+1 numbers with *a* < *b*<sup>*n*</sup>, *LLR* will do the *N*−1 Pocklington primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) and can prove that these numbers are primes, also for *a*×*b*<sup>*n*</sup>−1 numbers *a* < *b*<sup>*n*</sup>, *LLR* will do the *N*+1 Morrison algorithm (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) and can prove that these numbers are primes) → trial factor (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) from 10<sup>9</sup> to 10<sup>16</sup> → use *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to elliptic curve primality prove (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) the numbers < 10<sup>25000</sup>.

We are unable to determine if the families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) contain a prime (only count the numbers > *b*) or not (even if we allow strong probable primes), i.e. these families have no known prime (or strong probable prime) members > *b*, nor can they be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), i.e. whether these families contain a prime or a strong probable prime (only count the numbers > *b*) are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and all of these families are expected to contain a prime > *b* (in fact, expected to contain infinitely many primes), since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for *s* = 509203 and *s* = −78557, *s* = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and *s* = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://mersenneforum.org/showpost.php?p=564786&postcount=3, https://mersenneforum.org/showpost.php?p=461665&postcount=7, https://mersenneforum.org/showpost.php?p=625978&postcount=1027), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) *n*-digit base *b* number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/*n* (more accurately, the chance is approximately 1/(*n*×*ln*(*b*)), where *ln* is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base *e* = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113)). If one conjectures the numbers *x*{*y*}*z* behave similarly (i.e. the numbers *x*{*y*}*z* is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form *x*{*y*}*z* (of course, this does not always happen, since some *x*{*y*}*z* families can be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=421186&postcount=19, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://mersenneforum.org/showpost.php?p=564786&postcount=3, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), but it is at least a reasonable conjecture in the absence of evidence to the contrary).

Some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A, this family has been searched to *n* = 500000 with no prime or probable prime found, we can use ">500000" for the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A (while for the *n* of the smallest prime in the base 13 family 95<sub>*n*</sub>, it is 197420), ">500000" includes infinity (since infinity is > 500000) but does not includes 0 or −1, it is still possible that there is no prime in the base 13 family A3<sub>*n*</sub>A, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set *S* of the numbers *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, then this set *S* is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest *n* ≥ 1 such that *k*×2<sup>*n*</sup>+1 is prime" is ∞ for *k* = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for *k* = 10223 and 13018586 for *k* = 19249, another example is "the smallest *n* such that (*b*<sup>*n*</sup>−1)/(*b*−1) is prime" is ∞ for *b* = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for *b* = 691 and 41189 for *b* = 693).

There are also unproven probable primes (however, in this project our results assume that they are in fact primes, since they are > 10<sup>25000</sup> and the probability that they are in fact composite is < 10<sup>−2000</sup>, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf)), the unproven probable primes for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 6 unsolved families) are (together with the factorization of the numbers in their corresponding families):

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and http://alfredreichlg.de/, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm)

(all small prime factors (< 10<sup>9</sup>, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) of these probable primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these probable primes were added to *factordb*, unfortunely, none of these numbers have algebraic factors)

|*b*|index of this minimal prime in base *b* (assuming the primality of all probable primes in base *b*)|base-*b* form of this unproven probable prime (which is a minimal prime assuming its primality)|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this unproven probable prime (which is a minimal prime assuming its primality)|length of this unproven probable prime (which is a minimal prime assuming its primality) written in base *b*|length of this unproven probable prime (which is a minimal prime assuming its primality) written in decimal|*factordb* entry of this unproven probable prime (which is a minimal prime assuming its primality)|this unproven probable prime (which is a minimal prime assuming its primality) written in base *b*|this unproven probable prime (which is a minimal prime assuming its primality) written in decimal|*Primo* input file of this unproven probable prime (which is a minimal prime assuming its primality)|factorization of the numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|---|---|---|---|---|---|
|11|1068|57<sub>62668</sub>|(57×11<sup>62668</sup>−7)/10|62669|65263|http://factordb.com/index.php?id=1100000003573679860&open=prime|http://factordb.com/index.php?showid=1100000003573679860&base=11|http://factordb.com/index.php?showid=1100000003573679860|http://factordb.com/cert.php?id=1100000003573679860&inputfile|http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3194|C5<sub>23755</sub>C|(149×13<sup>23756</sup>+79)/12|23757|26464|http://factordb.com/index.php?id=1100000003590647776&open=prime|http://factordb.com/index.php?showid=1100000003590647776&base=13|http://factordb.com/index.php?showid=1100000003590647776|http://factordb.com/cert.php?id=1100000003590647776&inputfile|http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3195|80<sub>32017</sub>111|8×13<sup>32020</sup>+183|32021|35670|http://factordb.com/index.php?id=1100000000490878060&open=prime|http://factordb.com/index.php?showid=1100000000490878060&base=13|http://factordb.com/index.php?showid=1100000000490878060|http://factordb.com/cert.php?id=1100000000490878060&inputfile|http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3196|95<sub>197420</sub>|(113×13<sup>197420</sup>−5)/12|197421|219916|http://factordb.com/index.php?id=1100000003943359311&open=prime|http://factordb.com/index.php?showid=1100000003943359311&base=13|http://factordb.com/index.php?showid=1100000003943359311|(no *Primo* input file, since this unproven probable prime is too large (> 10<sup>149999</sup>) to be PRP-tested in *factordb*, and *factordb* does not have *Primo* input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has *Primo* input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))|http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2345|DB<sub>32234</sub>|(206×16<sup>32234</sup>−11)/15|32235|38815|http://factordb.com/index.php?id=1100000002383583629&open=prime|http://factordb.com/index.php?showid=1100000002383583629&base=16|http://factordb.com/index.php?showid=1100000002383583629|http://factordb.com/cert.php?id=1100000002383583629&inputfile|http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2346|4<sub>72785</sub>DD|(4×16<sup>72787</sup>+2291)/15|72787|87644|http://factordb.com/index.php?id=1100000003615909841&open=prime|http://factordb.com/index.php?showid=1100000003615909841&base=16|http://factordb.com/index.php?showid=1100000003615909841|http://factordb.com/cert.php?id=1100000003615909841&inputfile|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2347|3<sub>116137</sub>AF|(16<sup>116139</sup>+619)/5|116139|139845|http://factordb.com/index.php?id=1100000003851731988&open=prime|http://factordb.com/index.php?showid=1100000003851731988&base=16|http://factordb.com/index.php?showid=1100000003851731988|http://factordb.com/cert.php?id=1100000003851731988&inputfile|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|8003|BK<sub>22001</sub>5|(251×22<sup>22002</sup>−335)/21|22003|29538|http://factordb.com/index.php?id=1100000003594696838&open=prime|http://factordb.com/index.php?showid=1100000003594696838&base=22|http://factordb.com/index.php?showid=1100000003594696838|http://factordb.com/cert.php?id=1100000003594696838&inputfile|http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25250|5<sub>19391</sub>6F|(26<sup>19393</sup>+179)/5|19393|27440|http://factordb.com/index.php?id=1100000003850151202&open=prime|http://factordb.com/index.php?showid=1100000003850151202&base=26|http://factordb.com/index.php?showid=1100000003850151202|http://factordb.com/cert.php?id=1100000003850151202&inputfile|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25251|7<sub>20279</sub>OL|(7×26<sup>20281</sup>+11393)/25|20281|28697|http://factordb.com/index.php?id=1100000003892628605&open=prime|http://factordb.com/index.php?showid=1100000003892628605&base=26|http://factordb.com/index.php?showid=1100000003892628605|http://factordb.com/cert.php?id=1100000003892628605&inputfile|http://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25252|LD0<sub>20975</sub>7|559×26<sup>20976</sup>+7|20978|29684|http://factordb.com/index.php?id=1100000003892628658&open=prime|http://factordb.com/index.php?showid=1100000003892628658&base=26|http://factordb.com/index.php?showid=1100000003892628658|http://factordb.com/cert.php?id=1100000003892628658&inputfile|http://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25253|6K<sub>23300</sub>5|(34×26<sup>23301</sup>−79)/5|23302|32972|http://factordb.com/index.php?id=1100000003892628745&open=prime|http://factordb.com/index.php?showid=1100000003892628745&base=26|http://factordb.com/index.php?showid=1100000003892628745|http://factordb.com/cert.php?id=1100000003892628745&inputfile|http://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25254|J0<sub>44303</sub>KCB|19×26<sup>44306</sup>+13843|44307|62694|http://factordb.com/index.php?id=1100000003968156595&open=prime|http://factordb.com/index.php?showid=1100000003968156595&base=26|http://factordb.com/index.php?showid=1100000003968156595|http://factordb.com/cert.php?id=1100000003968156595&inputfile|http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B13843&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25255|M0<sub>61186</sub>2BB|22×26<sup>61189</sup>+1649|61190|86583|http://factordb.com/index.php?id=1100000003968169875&open=prime|http://factordb.com/index.php?showid=1100000003968169875&base=26|http://factordb.com/index.php?showid=1100000003968169875|http://factordb.com/cert.php?id=1100000003968169875&inputfile|http://factordb.com/index.php?query=22*26%5E%28n%2B3%29%2B1649&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25526|N6<sub>24051</sub>LR|(209×28<sup>24053</sup>+3967)/9|24054|34810|http://factordb.com/index.php?id=1100000003879667576&open=prime|http://factordb.com/index.php?showid=1100000003879667576&base=28|http://factordb.com/index.php?showid=1100000003879667576|http://factordb.com/cert.php?id=1100000003879667576&inputfile|http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25527|5OA<sub>31238</sub>F|(4438×28<sup>31239</sup>+125)/27|31241|45210|http://factordb.com/index.php?id=1100000003880455200&open=prime|http://factordb.com/index.php?showid=1100000003880455200&base=28|http://factordb.com/index.php?showid=1100000003880455200|http://factordb.com/cert.php?id=1100000003880455200&inputfile|http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25528|O4O<sub>94535</sub>9|(6092×28<sup>94536</sup>−143)/9|94538|136812|http://factordb.com/index.php?id=1100000000808118231&open=prime|http://factordb.com/index.php?showid=1100000000808118231&base=28|http://factordb.com/index.php?showid=1100000000808118231|http://factordb.com/cert.php?id=1100000000808118231&inputfile|http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2618|I0<sub>24608</sub>D|18×30<sup>24609</sup>+13|24610|36352|http://factordb.com/index.php?id=1100000003593967511&open=prime|http://factordb.com/index.php?showid=1100000003593967511&base=30|http://factordb.com/index.php?showid=1100000003593967511|http://factordb.com/cert.php?id=1100000003593967511&inputfile|http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35284|7K<sub>26567</sub>Z|(53×36<sup>26568</sup>+101)/7|26569|41349|http://factordb.com/index.php?id=1100000003896952461&open=prime|http://factordb.com/index.php?showid=1100000003896952461&base=36|http://factordb.com/index.php?showid=1100000003896952461|http://factordb.com/cert.php?id=1100000003896952461&inputfile|http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35285|S0<sub>75007</sub>8H|28×36<sup>75009</sup>+305|75010|116739|http://factordb.com/index.php?id=1100000004020085177&open=prime|http://factordb.com/index.php?showid=1100000004020085177&base=36|http://factordb.com/index.php?showid=1100000004020085177|http://factordb.com/cert.php?id=1100000004020085177&inputfile|http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35286|P<sub>81993</sub>SZ|(5×36<sup>81995</sup>+821)/7|81995|127609|http://factordb.com/index.php?id=1100000002394962083&open=prime|http://factordb.com/index.php?showid=1100000002394962083&base=36|http://factordb.com/index.php?showid=1100000002394962083|http://factordb.com/cert.php?id=1100000002394962083&inputfile|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

All these numbers are strong probable primes (https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/StrongPRP.html, https://mathworld.wolfram.com/StrongPseudoprime.html) to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 (see https://oeis.org/A014233 and https://oeis.org/A141768 and https://oeis.org/A001262 and https://oeis.org/A074773 and http://ntheory.org/data/spsps.txt), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (*P*, *Q*) defined by Selfridge's Method *A* (see https://oeis.org/A217255 and http://ntheory.org/data/slpsps-baillie.txt), and trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) to 10<sup>16</sup> (i.e. all these numbers are 10<sup>16</sup>-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html), and no composites < 2<sup>64</sup> pass the Baillie–PSW probable prime test (see http://ntheory.org/pseudoprimes.html and https://faculty.lynchburg.edu/~nicely/misc/bpsw.html), thus if one of these numbers is in fact composite, it will be a pseudoprime to the Baillie–PSW probable prime test, which currently no single example is known!

The unsolved families for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 6 unsolved families) and the factorization of the numbers in these families:

(you can calculate "equivalent searching limit of length in decimal" by: "current searching limit of length of this family" × *log*(*b*), where *log* is the common logarithm (https://en.wikipedia.org/wiki/Common_logarithm, https://mathworld.wolfram.com/CommonLogarithm.html), i.e. the logarithm with base 10)

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and http://alfredreichlg.de/, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm)

(all small prime factors (< 10<sup>9</sup>, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in these unsolved families were added to *factordb*, unfortunely, none of these numbers have algebraic factors)

|*b*|base-*b* form of the unsolved family|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the unsolved family|current searching limit of length of this family|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|
|13|A{3}A|(41×13<sup>*n*+1</sup>+27)/4|500000|http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|85{M}B|(5347×26<sup>*n*+1</sup>−297)/25|100000|http://factordb.com/index.php?query=%285347*26%5E%28n%2B1%29-297%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|{A}6F|(2×26<sup>*n*+2</sup>−497)/5|100000|http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29-497%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|{H}MH|(17×26<sup>*n*+2</sup>+3233)/25|100000|http://factordb.com/index.php?query=%2817*26%5E%28n%2B2%29%2B3233%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|{I}GL|(18×26<sup>*n*+2</sup>−1243)/25|100000|http://factordb.com/index.php?query=%2818*26%5E%28n%2B2%29-1243%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|O{A}F|(658×28<sup>*n*+1</sup>+125)/27|543203|http://factordb.com/index.php?query=(658*28^(n%2B1)%2B125)/27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|B{0}EUV|11×36<sup>*n*+3</sup>+19255|100000|http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|HM{0}N|634×36<sup>*n*+1</sup>+23|100000|http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|N{0}YYN|23×36<sup>*n*+3</sup>+45311|100000|http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|O{L}Z|(123×36<sup>*n*+1</sup>+67)/5|100000|http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

The large proven primes (> 10<sup>299</sup>) for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 6 unsolved families) and their primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php) and the factorization of the numbers in their corresponding families:

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and http://alfredreichlg.de/, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm)

(all small prime factors (< 10<sup>9</sup>, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) of these primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these primes were added to *factordb*, unfortunely, the only numbers having algebraic factors (other than trivially fully factored (i.e. primes of the form *k*×*b*<sup>*n*</sup>±1, with small *k*) and Cunningham numbers (of the form *b*<sup>*n*</sup>±1, see https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, https://www.numbersaplenty.com/set/Cunningham_number/, https://en.wikipedia.org/wiki/Cunningham_Project, https://t5k.org/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)), which are in the "README" file of the "primality-certificates" folder: https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md) are 8×18<sup>299</sup>+12 (the *N*+1 of the 547th minimal prime in base *b* = 18, 80<sub>298</sub>B = 8×18<sup>299</sup>+11) (which has sum-of-two-cubes factorization and can be factored to 12 × (6×18<sup>99</sup>+1) × (2×18<sup>199</sup>−6×18<sup>99</sup>+1)) and 18<sup>768</sup>−36 (the *N*+1 of the 548th minimal prime in base *b* = 18, H<sub>766</sub>FH = 18<sup>768</sup>−37) (which has difference-of-two-squares factorization and can be factored to (18<sup>384</sup>−6) × (18<sup>384</sup>+6)) and (16×20<sup>*n*+2</sup>−2809)/19 (the corresponding family of the 3307th minimal prime in base *b* = 20, G<sub>447</sub>99 = (16×20<sup>449</sup>−2809)/19) with even *n* (which has difference-of-two-squares factorization and can be factored to (4<sup>(*n*+2)/2</sup>−53) × (4<sup>(*n*+2)/2</sup>+53) / 19) and (22<sup>*n*+2</sup>−289)/3 (the corresponding family of the 8002nd minimal prime in base *b* = 22, 7<sub>3815</sub>2L = (22<sup>3817</sup>−289)/3) with even *n* (which has difference-of-two-squares factorization and can be factored to (22<sup>(*n*+2)/2</sup>−17) × (22<sup>(*n*+2)/2</sup>+17) / 3) and 25×30<sup>*n*</sup>−1 (the corresponding family of the 2619th minimal prime in base *b* = 30, OT<sub>34205</sub> = 25×30<sup>34205</sup>−1) with even *n* (which has difference-of-two-squares factorization and can be factored to (5×30<sup>*n*/2</sup>−1) × (5×30<sup>*n*/2</sup>+1))) (of course, 13<sup>*n*</sup>−49 (the *N*+1 of the 3193rd minimal prime in base *b* = 13, C<sub>10631</sub>92 = 13<sup>10633</sup>−50, is 13<sup>10633</sup>−49) has difference-of-two-squares factorization (factored to (13<sup>*n*/2</sup>−7) × (13<sup>*n*/2</sup>+7)) if *n* is even, but 10633 is odd) (of course, 22<sup>*n*</sup>−128 (the *N*+1 of the 7995th minimal prime in base *b* = 22, L<sub>483</sub>G3 = 22<sup>485</sup>−129, is 22<sup>485</sup>−128) has difference-of-two-7th-powers factorization (factored to (22<sup>*n*/7</sup>−2) × (22<sup>6×*n*/7</sup>+2×22<sup>5×*n*/7</sup>+4×22<sup>4×*n*/7</sup>+8×22<sup>3×*n*/7</sup>+16×22<sup>2×*n*/7</sup>+32×22<sup>*n*/7</sup>+64)) if *n* is divisible by 7, but 485 is not divisible by 7) (of course, 2×24<sup>*n*</sup>+6 (the *N*−1 of the 3403rd minimal prime in base *b* = 24, 20<sub>313</sub>7 = 2×24<sup>314</sup>+7, is 2×24<sup>314</sup>+6) has sum-of-two-cubes factorization (factored to 6 × (2×24<sup>(*n*−1)/3</sup>+1) × (4×24<sup>2×(*n*−1)/3</sup>−2×24<sup>(*n*−1)/3</sup>+1)) if *n* == 1 mod 3, but 314 is not == 1 mod 3) (of course, 2×24<sup>*n*</sup>+8 (the *N*+1 of the 3403rd minimal prime in base *b* = 24, 20<sub>313</sub>7 = 2×24<sup>314</sup>+7, is 2×24<sup>314</sup>+8) has Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup> (factored to 2 × (24<sup>*n*/2</sup>−2×<sup>*n*/4</sup>+2) × (24<sup>*n*/2</sup>+2×<sup>*n*/4</sup>+2)) if *n* is divisible by 4, but 314 is not divisible by 4) (of course, 4×13<sup>*n*</sup>+1 has Aurifeuillean factorization of *x*<sup>4</sup>+4×*y*<sup>4</sup> (factored to (2×13<sup>*n*/2</sup>−2×13<sup>*n*/4</sup>+1) × (2×13<sup>*n*/2</sup>+2×13<sup>*n*/4</sup>+1)) if *n* is divisible by 4, but 16×13<sup>*n*</sup>+1 (which is a factor of 128×13<sup>*n*</sup>+8, the *N*−1 of the 3173rd minimal prime in base *b* = 13, 9B0<sub>391</sub>9 = 128×13<sup>392</sup>+9, is 128×13<sup>392</sup>+8) has no algebraic factors for any *n*)

|*b*|index of this minimal prime in base *b*|base-*b* form of this minimal prime|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this minimal prime|length of this minimal prime written in base *b*|length of this minimal prime written in decimal|*factordb* entry of this minimal prime|this minimal prime written in base *b*|this minimal prime written in decimal|primality certificate for this minimal prime|factorization of the numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|
|---|---|---|---|---|---|---|---|---|---|---|
|9|149|76<sub>329</sub>2|(31×9<sup>330</sup>−19)/4|331|316|http://factordb.com/index.php?id=1100000002359003642&open=prime|http://factordb.com/index.php?showid=1100000002359003642&base=9|http://factordb.com/index.php?showid=1100000002359003642|http://factordb.com/cert.php?id=1100000002359003642|http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|150|27<sub>686</sub>07|(23×9<sup>688</sup>−511)/8|689|657|http://factordb.com/index.php?id=1100000002495467486&open=prime|http://factordb.com/index.php?showid=1100000002495467486&base=9|http://factordb.com/index.php?showid=1100000002495467486|http://factordb.com/cert.php?id=1100000002495467486|http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|9|151|30<sub>1158</sub>11|3×9<sup>1160</sup>+10|1161|1108|http://factordb.com/index.php?id=1100000002376318423&open=prime|http://factordb.com/index.php?showid=1100000002376318423&base=9|http://factordb.com/index.php?showid=1100000002376318423|http://factordb.com/cert.php?id=1100000002376318423|http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|1065|A<sub>713</sub>58|11<sup>715</sup>−58|715|745|http://factordb.com/index.php?id=1100000003576826487&open=prime|http://factordb.com/index.php?showid=1100000003576826487&base=11|http://factordb.com/index.php?showid=1100000003576826487|http://factordb.com/cert.php?id=1100000003576826487|http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|1066|7<sub>759</sub>44|(7×11<sup>761</sup>−367)/10|761|793|http://factordb.com/index.php?id=1100000002505568840&open=prime|http://factordb.com/index.php?showid=1100000002505568840&base=11|http://factordb.com/index.php?showid=1100000002505568840|http://factordb.com/cert.php?id=1100000002505568840|http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|11|1067|557<sub>1011</sub>|(607×11<sup>1011</sup>−7)/10|1013|1055|http://factordb.com/index.php?id=1100000002361376522&open=prime|http://factordb.com/index.php?showid=1100000002361376522&base=11|http://factordb.com/index.php?showid=1100000002361376522|http://factordb.com/cert.php?id=1100000002361376522|http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3165|50<sub>270</sub>44|5×13<sup>272</sup>+56|273|304|http://factordb.com/index.php?id=1100000002632397005&open=prime|http://factordb.com/index.php?showid=1100000002632397005&base=13|http://factordb.com/index.php?showid=1100000002632397005|http://factordb.com/cert.php?id=1100000002632397005|http://factordb.com/index.php?query=5*13%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3166|9<sub>271</sub>095|(3×13<sup>274</sup>−6103)/4|274|306|http://factordb.com/index.php?id=1100000003590431654&open=prime|http://factordb.com/index.php?showid=1100000003590431654&base=13|http://factordb.com/index.php?showid=1100000003590431654|http://factordb.com/cert.php?id=1100000003590431654|http://factordb.com/index.php?query=%283*13%5E%28n%2B3%29-6103%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3167|10<sub>286</sub>7771|13<sup>290</sup>+16654|291|324|http://factordb.com/index.php?id=1100000003590431633&open=prime|http://factordb.com/index.php?showid=1100000003590431633&base=13|http://factordb.com/index.php?showid=1100000003590431633|http://factordb.com/cert.php?id=1100000003590431633|http://factordb.com/index.php?query=13%5E%28n%2B4%29%2B16654&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3168|9<sub>308</sub>1|(3×13<sup>309</sup>−35)/4|309|345|http://factordb.com/index.php?id=1100000000840126705&open=prime|http://factordb.com/index.php?showid=1100000000840126705&base=13|http://factordb.com/index.php?showid=1100000000840126705|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 39/4×(13<sup>308</sup>−1), thus factor *N*−1 is equivalent to factor 13<sup>308</sup>−1, and for the factorization of 13<sup>308</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-35%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3169|B<sub>341</sub>C4|(11×13<sup>343</sup>+61)/12|343|383|http://factordb.com/index.php?id=1100000003590431618&open=prime|http://factordb.com/index.php?showid=1100000003590431618&base=13|http://factordb.com/index.php?showid=1100000003590431618|http://factordb.com/cert.php?id=1100000003590431618|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29%2B61%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3170|8B<sub>343</sub>|(107×13<sup>343</sup>−11)/12|344|384|http://factordb.com/index.php?id=1100000002321018736&open=prime|http://factordb.com/index.php?showid=1100000002321018736&base=13|http://factordb.com/index.php?showid=1100000002321018736|http://factordb.com/cert.php?id=1100000002321018736|http://factordb.com/index.php?query=%28107*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3171|710<sub>371</sub>111|92×13<sup>374</sup>+183|376|419|http://factordb.com/index.php?id=1100000003590431609&open=prime|http://factordb.com/index.php?showid=1100000003590431609&base=13|http://factordb.com/index.php?showid=1100000003590431609|http://factordb.com/cert.php?id=1100000003590431609|http://factordb.com/index.php?query=92*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3172|75<sub>375</sub>7|(89×13<sup>376</sup>+19)/12|377|420|http://factordb.com/index.php?id=1100000003590431596&open=prime|http://factordb.com/index.php?showid=1100000003590431596&base=13|http://factordb.com/index.php?showid=1100000003590431596|http://factordb.com/cert.php?id=1100000003590431596|http://factordb.com/index.php?query=%2889*13%5E%28n%2B1%29%2B19%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3173|9B0<sub>391</sub>9|128×13<sup>392</sup>+9|394|439|http://factordb.com/index.php?id=1100000002632396790&open=prime|http://factordb.com/index.php?showid=1100000002632396790&base=13|http://factordb.com/index.php?showid=1100000002632396790|http://factordb.com/cert.php?id=1100000002632396790|http://factordb.com/index.php?query=128*13%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3174|7B0B<sub>397</sub>|(15923×13<sup>397</sup>−11)/12|400|446|http://factordb.com/index.php?id=1100000003590431574&open=prime|http://factordb.com/index.php?showid=1100000003590431574&base=13|http://factordb.com/index.php?showid=1100000003590431574|http://factordb.com/cert.php?id=1100000003590431574|http://factordb.com/index.php?query=%2815923*13%5En-11%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3175|10<sub>414</sub>93|13<sup>416</sup>+120|417|464|http://factordb.com/index.php?id=1100000002523249240&open=prime|http://factordb.com/index.php?showid=1100000002523249240&base=13|http://factordb.com/index.php?showid=1100000002523249240|http://factordb.com/cert.php?id=1100000002523249240|http://factordb.com/index.php?query=13%5E%28n%2B2%29%2B120&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3176|81010<sub>415</sub>1|17746×13<sup>416</sup>+1|420|468|http://factordb.com/index.php?id=1100000003590431555&open=prime|http://factordb.com/index.php?showid=1100000003590431555&base=13|http://factordb.com/index.php?showid=1100000003590431555|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=17746*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3177|8110<sub>435</sub>1|1366×13<sup>436</sup>+1|439|489|http://factordb.com/index.php?id=1100000002373259109&open=prime|http://factordb.com/index.php?showid=1100000002373259109&base=13|http://factordb.com/index.php?showid=1100000002373259109|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=1366*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3178|B7<sub>486</sub>|(139×13<sup>486</sup>−7)/12|487|543|http://factordb.com/index.php?id=1100000002321015892&open=prime|http://factordb.com/index.php?showid=1100000002321015892&base=13|http://factordb.com/index.php?showid=1100000002321015892|http://factordb.com/cert.php?id=1100000002321015892|http://factordb.com/index.php?query=%28139*13%5En-7%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3179|B<sub>563</sub>C|(11×13<sup>564</sup>+1)/12|564|629|http://factordb.com/index.php?id=1100000000000217927&open=prime|http://factordb.com/index.php?showid=1100000000000217927&base=13|http://factordb.com/index.php?showid=1100000000000217927|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 11/12×(13<sup>564</sup>−1), thus factor *N*−1 is equivalent to factor 13<sup>564</sup>−1, and for the factorization of 13<sup>564</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=|http://factordb.com/index.php?query=%2811*13%5E%28n%2B1%29%2B1%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3180|1B<sub>576</sub>|(23×13<sup>576</sup>−11)/12|577|642|http://factordb.com/index.php?id=1100000002321021456&open=prime|http://factordb.com/index.php?showid=1100000002321021456&base=13|http://factordb.com/index.php?showid=1100000002321021456|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 23/12×(13<sup>576</sup>−1), thus factor *N*−1 is equivalent to factor 13<sup>576</sup>−1, and for the factorization of 13<sup>576</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=|http://factordb.com/index.php?query=%2823*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3181|80<sub>693</sub>87|8×13<sup>695</sup>+111|696|776|http://factordb.com/index.php?id=1100000002615636527&open=prime|http://factordb.com/index.php?showid=1100000002615636527&base=13|http://factordb.com/index.php?showid=1100000002615636527|http://factordb.com/cert.php?id=1100000002615636527|http://factordb.com/index.php?query=8*13%5E%28n%2B2%29%2B111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3182|CC5<sub>713</sub>|(2021×13<sup>713</sup>−5)/12|715|797|http://factordb.com/index.php?id=1100000002615627353&open=prime|http://factordb.com/index.php?showid=1100000002615627353&base=13|http://factordb.com/index.php?showid=1100000002615627353|http://factordb.com/cert.php?id=1100000002615627353|http://factordb.com/index.php?query=%282021*13%5En-5%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3183|B<sub>834</sub>74|(11×13<sup>836</sup>−719)/12|836|932|http://factordb.com/index.php?id=1100000003590430871&open=prime|http://factordb.com/index.php?showid=1100000003590430871&base=13|http://factordb.com/index.php?showid=1100000003590430871|http://factordb.com/cert.php?id=1100000003590430871|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29-719%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3184|9<sub>968</sub>B|(3×13<sup>969</sup>+5)/4|969|1080|http://factordb.com/index.php?id=1100000000258566244&open=prime|http://factordb.com/index.php?showid=1100000000258566244&base=13|http://factordb.com/index.php?showid=1100000000258566244|http://factordb.com/cert.php?id=1100000000258566244|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29%2B5%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3185|10<sub>1295</sub>181|13<sup>1298</sup>+274|1299|1446|http://factordb.com/index.php?id=1100000002615445013&open=prime|http://factordb.com/index.php?showid=1100000002615445013&base=13|http://factordb.com/index.php?showid=1100000002615445013|http://factordb.com/cert.php?id=1100000002615445013|http://factordb.com/index.php?query=13%5E%28n%2B3%29%2B274&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3186|9<sub>1362</sub>5|(3×13<sup>1363</sup>−19)/4|1363|1519|http://factordb.com/index.php?id=1100000002321017776&open=prime|http://factordb.com/index.php?showid=1100000002321017776&base=13|http://factordb.com/index.php?showid=1100000002321017776|http://factordb.com/cert.php?id=1100000002321017776|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-19%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3187|7<sub>1504</sub>1|(7×13<sup>1505</sup>−79)/12|1505|1677|http://factordb.com/index.php?id=1100000002320890755&open=prime|http://factordb.com/index.php?showid=1100000002320890755&base=13|http://factordb.com/index.php?showid=1100000002320890755|http://factordb.com/cert.php?id=1100000002320890755|http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29-79%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3188|930<sub>1551</sub>1|120×13<sup>1552</sup>+1|1554|1731|http://factordb.com/index.php?id=1100000000765961452&open=prime|http://factordb.com/index.php?showid=1100000000765961452&base=13|http://factordb.com/index.php?showid=1100000000765961452|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3189|720<sub>2297</sub>2|93×13<sup>2298</sup>+2|2300|2562|http://factordb.com/index.php?id=1100000002632396910&open=prime|http://factordb.com/index.php?showid=1100000002632396910&base=13|http://factordb.com/index.php?showid=1100000002632396910|http://factordb.com/cert.php?id=1100000002632396910|http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3190|1770<sub>2703</sub>17|267×13<sup>2705</sup>+20|2708|3016|http://factordb.com/index.php?id=1100000003590430825&open=prime|http://factordb.com/index.php?showid=1100000003590430825&base=13|http://factordb.com/index.php?showid=1100000003590430825|http://factordb.com/cert.php?id=1100000003590430825|http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3191|390<sub>6266</sub>1|48×13<sup>6267</sup>+1|6269|6983|http://factordb.com/index.php?id=1100000000765961441&open=prime|http://factordb.com/index.php?showid=1100000000765961441&base=13|http://factordb.com/index.php?showid=1100000000765961441|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3192|B0<sub>6540</sub>BBA|11×13<sup>6543</sup>+2012|6544|7290|http://factordb.com/index.php?id=1100000002616382906&open=prime|http://factordb.com/index.php?showid=1100000002616382906&base=13|http://factordb.com/index.php?showid=1100000002616382906|http://factordb.com/cert.php?id=1100000002616382906|http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|13|3193|C<sub>10631</sub>92|13<sup>10633</sup>−50|10633|11845|http://factordb.com/index.php?id=1100000003590493750&open=prime|http://factordb.com/index.php?showid=1100000003590493750&base=13|http://factordb.com/index.php?showid=1100000003590493750|http://factordb.com/cert.php?id=1100000003590493750|http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|649|34D<sub>708</sub>|47×14<sup>708</sup>−1|710|814|http://factordb.com/index.php?id=1100000001540144903&open=prime|http://factordb.com/index.php?showid=1100000001540144903&base=14|http://factordb.com/index.php?showid=1100000001540144903|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|14|650|4D<sub>19698</sub>|5×14<sup>19698</sup>−1|19699|22578|http://factordb.com/index.php?id=1100000000884560233&open=prime|http://factordb.com/index.php?showid=1100000000884560233&base=14|http://factordb.com/index.php?showid=1100000000884560233|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2328|880<sub>246</sub>7|136×16<sup>247</sup>+7|249|300|http://factordb.com/index.php?id=1100000002468140199&open=prime|http://factordb.com/index.php?showid=1100000002468140199&base=16|http://factordb.com/index.php?showid=1100000002468140199|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2<sup>3</sup>×3×7×13×25703261×(289-digit prime)|http://factordb.com/index.php?query=136*16%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2329|D4<sub>263</sub>D|(199×16<sup>264</sup>+131)/15|265|320|http://factordb.com/index.php?id=1100000002468170238&open=prime|http://factordb.com/index.php?showid=1100000002468170238&base=16|http://factordb.com/index.php?showid=1100000002468170238|http://factordb.com/cert.php?id=1100000002468170238|http://factordb.com/index.php?query=%28199*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2330|E0<sub>261</sub>4DD|14×16<sup>264</sup>+1245|265|320|http://factordb.com/index.php?id=1100000003588388352&open=prime|http://factordb.com/index.php?showid=1100000003588388352&base=16|http://factordb.com/index.php?showid=1100000003588388352|http://factordb.com/cert.php?id=1100000003588388352|http://factordb.com/index.php?query=14*16%5E%28n%2B3%29%2B1245&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2331|8C0<sub>290</sub>ED|140×16<sup>292</sup>+237|294|354|http://factordb.com/index.php?id=1100000003588388307&open=prime|http://factordb.com/index.php?showid=1100000003588388307&base=16|http://factordb.com/index.php?showid=1100000003588388307|http://factordb.com/cert.php?id=1100000003588388307|http://factordb.com/index.php?query=140*16%5E%28n%2B2%29%2B237&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2332|DA<sub>305</sub>5|(41×16<sup>306</sup>−17)/3|307|370|http://factordb.com/index.php?id=1100000003588388284&open=prime|http://factordb.com/index.php?showid=1100000003588388284&base=16|http://factordb.com/index.php?showid=1100000003588388284|http://factordb.com/cert.php?id=1100000003588388284|http://factordb.com/index.php?query=%2841*16%5E%28n%2B1%29-17%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2333|CE80<sub>422</sub>D|3304×16<sup>423</sup>+13|426|513|http://factordb.com/index.php?id=1100000003588388257&open=prime|http://factordb.com/index.php?showid=1100000003588388257&base=16|http://factordb.com/index.php?showid=1100000003588388257|http://factordb.com/cert.php?id=1100000003588388257|http://factordb.com/index.php?query=3304*16%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2334|5F<sub>544</sub>6F|6×16<sup>546</sup>−145|547|659|http://factordb.com/index.php?id=1100000002604723967&open=prime|http://factordb.com/index.php?showid=1100000002604723967&base=16|http://factordb.com/index.php?showid=1100000002604723967|http://factordb.com/cert.php?id=1100000002604723967|http://factordb.com/index.php?query=6*16%5E%28n%2B2%29-145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2335|88F<sub>545</sub>|137×16<sup>545</sup>−1|547|659|http://factordb.com/index.php?id=1100000000413679658&open=prime|http://factordb.com/index.php?showid=1100000000413679658&base=16|http://factordb.com/index.php?showid=1100000000413679658|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=137*16%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2336|BE0<sub>792</sub>BB|190×16<sup>794</sup>+187|796|959|http://factordb.com/index.php?id=1100000003588387938&open=prime|http://factordb.com/index.php?showid=1100000003588387938&base=16|http://factordb.com/index.php?showid=1100000003588387938|http://factordb.com/cert.php?id=1100000003588387938|http://factordb.com/index.php?query=190*16%5E%28n%2B2%29%2B187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2337|D9<sub>1052</sub>|(68×16<sup>1052</sup>−3)/5|1053|1268|http://factordb.com/index.php?id=1100000002321036020&open=prime|http://factordb.com/index.php?showid=1100000002321036020&base=16|http://factordb.com/index.php?showid=1100000002321036020|http://factordb.com/cert.php?id=1100000002321036020|http://factordb.com/index.php?query=%2868*16%5En-3%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2338|FAF<sub>1062</sub>45|251×16<sup>1064</sup>−187|1066|1284|http://factordb.com/index.php?id=1100000003588387610&open=prime|http://factordb.com/index.php?showid=1100000003588387610&base=16|http://factordb.com/index.php?showid=1100000003588387610|http://factordb.com/cert.php?id=1100000003588387610|http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2339|F8<sub>1517</sub>F|(233×16<sup>1518</sup>+97)/15|1519|1830|http://factordb.com/index.php?id=1100000000633744824&open=prime|http://factordb.com/index.php?showid=1100000000633744824&base=16|http://factordb.com/index.php?showid=1100000000633744824|http://factordb.com/cert.php?id=1100000000633744824|http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2340|20<sub>1713</sub>321|2×16<sup>1716</sup>+801|1717|2067|http://factordb.com/index.php?id=1100000003588386735&open=prime|http://factordb.com/index.php?showid=1100000003588386735&base=16|http://factordb.com/index.php?showid=1100000003588386735|http://factordb.com/cert.php?id=1100000003588386735|http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2341|300F<sub>1960</sub>AF|769×16<sup>1962</sup>−81|1965|2366|http://factordb.com/index.php?id=1100000003588368750&open=prime|http://factordb.com/index.php?showid=1100000003588368750&base=16|http://factordb.com/index.php?showid=1100000003588368750|http://factordb.com/cert.php?id=1100000003588368750|http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2342|90<sub>3542</sub>91|9×16<sup>3544</sup>+145|3545|4269|http://factordb.com/index.php?id=1100000000633424191&open=prime|http://factordb.com/index.php?showid=1100000000633424191&base=16|http://factordb.com/index.php?showid=1100000000633424191|http://factordb.com/cert.php?id=1100000000633424191|http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2343|5BC<sub>3700</sub>D|(459×16<sup>3701</sup>+1)/5|3703|4459|http://factordb.com/index.php?id=1100000000993764322&open=prime|http://factordb.com/index.php?showid=1100000000993764322&base=16|http://factordb.com/index.php?showid=1100000000993764322|http://factordb.com/cert.php?id=1100000000993764322|http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|16|2344|D0B<sub>17804</sub>|(3131×16<sup>17804</sup>−11)/15|17806|21441|http://factordb.com/index.php?id=1100000003589278511&open=prime|http://factordb.com/index.php?showid=1100000003589278511&base=16|http://factordb.com/index.php?showid=1100000003589278511|http://factordb.com/cert.php?id=1100000003589278511|http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|18|547|80<sub>298</sub>B|8×18<sup>299</sup>+11|300|377|http://factordb.com/index.php?id=1100000002355574745&open=prime|http://factordb.com/index.php?showid=1100000002355574745&base=18|http://factordb.com/index.php?showid=1100000002355574745|http://factordb.com/cert.php?id=1100000002355574745|http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|18|548|H<sub>766</sub>FH|18<sup>768</sup>−37|768|965|http://factordb.com/index.php?id=1100000003590430490&open=prime|http://factordb.com/index.php?showid=1100000003590430490&base=18|http://factordb.com/index.php?showid=1100000003590430490|http://factordb.com/cert.php?id=1100000003590430490|http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|18|549|C0<sub>6268</sub>C5|12×18<sup>6270</sup>+221|6271|7872|http://factordb.com/index.php?id=1100000003590442437&open=prime|http://factordb.com/index.php?showid=1100000003590442437&base=18|http://factordb.com/index.php?showid=1100000003590442437|http://factordb.com/cert.php?id=1100000003590442437|http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3301|H<sub>247</sub>A0H|(17×20<sup>250</sup>−59677)/19|250|326|http://factordb.com/index.php?id=1100000003590502619&open=prime|http://factordb.com/index.php?showid=1100000003590502619&base=20|http://factordb.com/index.php?showid=1100000003590502619|http://factordb.com/cert.php?id=1100000003590502619|http://factordb.com/index.php?query=%2817*20%5E%28n%2B3%29-59677%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3302|7<sub>249</sub>A7|(7×20<sup>251</sup>+1133)/19|251|327|http://factordb.com/index.php?id=1100000003590502602&open=prime|http://factordb.com/index.php?showid=1100000003590502602&base=20|http://factordb.com/index.php?showid=1100000003590502602|http://factordb.com/cert.php?id=1100000003590502602|http://factordb.com/index.php?query=%287*20%5E%28n%2B2%29%2B1133%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3303|J7<sub>270</sub>|(368×20<sup>270</sup>−7)/19|271|353|http://factordb.com/index.php?id=1100000002325395462&open=prime|http://factordb.com/index.php?showid=1100000002325395462&base=20|http://factordb.com/index.php?showid=1100000002325395462|http://factordb.com/cert.php?id=1100000002325395462|http://factordb.com/index.php?query=%28368*20%5En-7%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3304|J<sub>330</sub>CCC7|20<sup>334</sup>−58953|334|435|http://factordb.com/index.php?id=1100000003590502572&open=prime|http://factordb.com/index.php?showid=1100000003590502572&base=20|http://factordb.com/index.php?showid=1100000003590502572|http://factordb.com/cert.php?id=1100000003590502572|http://factordb.com/index.php?query=20%5E%28n%2B4%29-58953&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3305|40<sub>387</sub>404B|4×20<sup>391</sup>+32091|392|510|http://factordb.com/index.php?id=1100000003590502563&open=prime|http://factordb.com/index.php?showid=1100000003590502563&base=20|http://factordb.com/index.php?showid=1100000003590502563|http://factordb.com/cert.php?id=1100000003590502563|http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3306|EC0<sub>429</sub>7|292×20<sup>430</sup>+7|432|562|http://factordb.com/index.php?id=1100000002633348702&open=prime|http://factordb.com/index.php?showid=1100000002633348702&base=20|http://factordb.com/index.php?showid=1100000002633348702|http://factordb.com/cert.php?id=1100000002633348702|http://factordb.com/index.php?query=292*20%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3307|G<sub>447</sub>99|(16×20<sup>449</sup>−2809)/19|449|585|http://factordb.com/index.php?id=1100000000840126753&open=prime|http://factordb.com/index.php?showid=1100000000840126753&base=20|http://factordb.com/index.php?showid=1100000000840126753|http://factordb.com/cert.php?id=1100000000840126753|http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3308|3A<sub>527</sub>3|(67×20<sup>528</sup>−143)/19|529|688|http://factordb.com/index.php?id=1100000003590502531&open=prime|http://factordb.com/index.php?showid=1100000003590502531&base=20|http://factordb.com/index.php?showid=1100000003590502531|http://factordb.com/cert.php?id=1100000003590502531|http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3309|E<sub>566</sub>C7|(14×20<sup>568</sup>−907)/19|568|739|http://factordb.com/index.php?id=1100000003590502516&open=prime|http://factordb.com/index.php?showid=1100000003590502516&base=20|http://factordb.com/index.php?showid=1100000003590502516|http://factordb.com/cert.php?id=1100000003590502516|http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3310|JCJ<sub>629</sub>|393×20<sup>629</sup>−1|631|821|http://factordb.com/index.php?id=1100000001559454258&open=prime|http://factordb.com/index.php?showid=1100000001559454258&base=20|http://factordb.com/index.php?showid=1100000001559454258|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3311|J<sub>655</sub>05J|20<sup>658</sup>−7881|658|857|http://factordb.com/index.php?id=1100000003590502490&open=prime|http://factordb.com/index.php?showid=1100000003590502490&base=20|http://factordb.com/index.php?showid=1100000003590502490|http://factordb.com/cert.php?id=1100000003590502490|http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3312|50<sub>1163</sub>AJ|5×20<sup>1165</sup>+219|1166|1517|http://factordb.com/index.php?id=1100000003590502412&open=prime|http://factordb.com/index.php?showid=1100000003590502412&base=20|http://factordb.com/index.php?showid=1100000003590502412|http://factordb.com/cert.php?id=1100000003590502412|http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3313|CD<sub>2449</sub>|(241×20<sup>2449</sup>−13)/19|2450|3188|http://factordb.com/index.php?id=1100000002325393915&open=prime|http://factordb.com/index.php?showid=1100000002325393915&base=20|http://factordb.com/index.php?showid=1100000002325393915|http://factordb.com/cert.php?id=1100000002325393915|http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|20|3314|G0<sub>6269</sub>D|16×20<sup>6270</sup>+13|6271|8159|http://factordb.com/index.php?id=1100000003590539457&open=prime|http://factordb.com/index.php?showid=1100000003590539457&base=20|http://factordb.com/index.php?showid=1100000003590539457|http://factordb.com/cert.php?id=1100000003590539457|http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7984|I7G0<sub>254</sub>H|8882×22<sup>255</sup>+17|258|347|http://factordb.com/index.php?id=1100000003591372788&open=prime|http://factordb.com/index.php?showid=1100000003591372788&base=22|http://factordb.com/index.php?showid=1100000003591372788|http://factordb.com/cert.php?id=1100000003591372788|http://factordb.com/index.php?query=8882*22%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7985|D0<sub>255</sub>5EEF|13×22<sup>259</sup>+60339|260|349|http://factordb.com/index.php?id=1100000003591371932&open=prime|http://factordb.com/index.php?showid=1100000003591371932&base=22|http://factordb.com/index.php?showid=1100000003591371932|http://factordb.com/cert.php?id=1100000003591371932|http://factordb.com/index.php?query=13*22%5E%28n%2B4%29%2B60339&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7986|IK<sub>322</sub>F|(398×22<sup>323</sup>−125)/21|324|435|http://factordb.com/index.php?id=1100000000840384145&open=prime|http://factordb.com/index.php?showid=1100000000840384145&base=22|http://factordb.com/index.php?showid=1100000000840384145|http://factordb.com/cert.php?id=1100000000840384145|http://factordb.com/index.php?query=%28398*22%5E%28n%2B1%29-125%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7987|C0<sub>340</sub>G9|12×22<sup>342</sup>+361|343|461|http://factordb.com/index.php?id=1100000000840384159&open=prime|http://factordb.com/index.php?showid=1100000000840384159&base=22|http://factordb.com/index.php?showid=1100000000840384159|http://factordb.com/cert.php?id=1100000000840384159|http://factordb.com/index.php?query=12*22%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7988|77E<sub>348</sub>K7|(485×22<sup>350</sup>+373)/3|352|473|http://factordb.com/index.php?id=1100000003591369779&open=prime|http://factordb.com/index.php?showid=1100000003591369779&base=22|http://factordb.com/index.php?showid=1100000003591369779|http://factordb.com/cert.php?id=1100000003591369779|http://factordb.com/index.php?query=%28485*22%5E%28n%2B2%29%2B373%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7989|J<sub>379</sub>KJ|(19×22<sup>381</sup>+443)/21|381|512|http://factordb.com/index.php?id=1100000003591369027&open=prime|http://factordb.com/index.php?showid=1100000003591369027&base=22|http://factordb.com/index.php?showid=1100000003591369027|http://factordb.com/cert.php?id=1100000003591369027|http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29%2B443%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7990|J<sub>388</sub>EJ|(19×22<sup>390</sup>−2329)/21|390|524|http://factordb.com/index.php?id=1100000003591367729&open=prime|http://factordb.com/index.php?showid=1100000003591367729&base=22|http://factordb.com/index.php?showid=1100000003591367729|http://factordb.com/cert.php?id=1100000003591367729|http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29-2329%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7991|DJ<sub>400</sub>|(292×22<sup>400</sup>−19)/21|401|539|http://factordb.com/index.php?id=1100000002325880110&open=prime|http://factordb.com/index.php?showid=1100000002325880110&base=22|http://factordb.com/index.php?showid=1100000002325880110|http://factordb.com/cert.php?id=1100000002325880110|http://factordb.com/index.php?query=%28292*22%5En-19%29%2F21&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7992|E<sub>404</sub>K7|(2×22<sup>406</sup>+373)/3|406|545|http://factordb.com/index.php?id=1100000003591366298&open=prime|http://factordb.com/index.php?showid=1100000003591366298&base=22|http://factordb.com/index.php?showid=1100000003591366298|http://factordb.com/cert.php?id=1100000003591366298|http://factordb.com/index.php?query=%282*22%5E%28n%2B2%29%2B373%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7993|66F<sub>453</sub>B3|(971×22<sup>455</sup>−705)/7|457|613|http://factordb.com/index.php?id=1100000003591365809&open=prime|http://factordb.com/index.php?showid=1100000003591365809&base=22|http://factordb.com/index.php?showid=1100000003591365809|http://factordb.com/cert.php?id=1100000003591365809|http://factordb.com/index.php?query=%28971*22%5E%28n%2B2%29-705%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7994|L0<sub>454</sub>B63|21×22<sup>457</sup>+5459|458|615|http://factordb.com/index.php?id=1100000003591365331&open=prime|http://factordb.com/index.php?showid=1100000003591365331&base=22|http://factordb.com/index.php?showid=1100000003591365331|http://factordb.com/cert.php?id=1100000003591365331|http://factordb.com/index.php?query=21*22%5E%28n%2B3%29%2B5459&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7995|L<sub>483</sub>G3|22<sup>485</sup>−129|485|652|http://factordb.com/index.php?id=1100000003591364730&open=prime|http://factordb.com/index.php?showid=1100000003591364730&base=22|http://factordb.com/index.php?showid=1100000003591364730|http://factordb.com/cert.php?id=1100000003591364730|http://factordb.com/index.php?query=22%5E%28n%2B2%29-129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7996|E60<sub>496</sub>L|314×22<sup>497</sup>+21|499|670|http://factordb.com/index.php?id=1100000000632703239&open=prime|http://factordb.com/index.php?showid=1100000000632703239&base=22|http://factordb.com/index.php?showid=1100000000632703239|http://factordb.com/cert.php?id=1100000000632703239|http://factordb.com/index.php?query=314*22%5E%28n%2B1%29%2B21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7997|I<sub>626</sub>AF|(6×22<sup>628</sup>−1259)/7|628|843|http://factordb.com/index.php?id=1100000000632724334&open=prime|http://factordb.com/index.php?showid=1100000000632724334&base=22|http://factordb.com/index.php?showid=1100000000632724334|http://factordb.com/cert.php?id=1100000000632724334|http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7998|K0<sub>760</sub>EC1|20×22<sup>763</sup>+7041|764|1026|http://factordb.com/index.php?id=1100000000632724415&open=prime|http://factordb.com/index.php?showid=1100000000632724415&base=22|http://factordb.com/index.php?showid=1100000000632724415|http://factordb.com/cert.php?id=1100000000632724415|http://factordb.com/index.php?query=20*22%5E%28n%2B3%29%2B7041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|7999|J0<sub>767</sub>IGGJ|19×22<sup>771</sup>+199779|772|1037|http://factordb.com/index.php?id=1100000003591362567&open=prime|http://factordb.com/index.php?showid=1100000003591362567&base=22|http://factordb.com/index.php?showid=1100000003591362567|http://factordb.com/cert.php?id=1100000003591362567|http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|8000|7<sub>959</sub>K7|(22<sup>961</sup>+857)/3|961|1290|http://factordb.com/index.php?id=1100000003591361817&open=prime|http://factordb.com/index.php?showid=1100000003591361817&base=22|http://factordb.com/index.php?showid=1100000003591361817|http://factordb.com/cert.php?id=1100000003591361817|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|8001|L<sub>2385</sub>KE7|22<sup>2388</sup>−653|2388|3206|http://factordb.com/index.php?id=1100000003591360774&open=prime|http://factordb.com/index.php?showid=1100000003591360774&base=22|http://factordb.com/index.php?showid=1100000003591360774|http://factordb.com/cert.php?id=1100000003591360774|http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|22|8002|7<sub>3815</sub>2L|(22<sup>3817</sup>−289)/3|3817|5124|http://factordb.com/index.php?id=1100000003591359839&open=prime|http://factordb.com/index.php?showid=1100000003591359839&base=22|http://factordb.com/index.php?showid=1100000003591359839|http://factordb.com/cert.php?id=1100000003591359839|http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3400|I0<sub>241</sub>I5|18×24<sup>243</sup>+437|244|337|http://factordb.com/index.php?id=1100000002633360037&open=prime|http://factordb.com/index.php?showid=1100000002633360037&base=24|http://factordb.com/index.php?showid=1100000002633360037|http://factordb.com/cert.php?id=1100000002633360037|http://factordb.com/index.php?query=18*24%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3401|D0<sub>259</sub>KKD|13×24<sup>262</sup>+12013|263|363|http://factordb.com/index.php?id=1100000003593270725&open=prime|http://factordb.com/index.php?showid=1100000003593270725&base=24|http://factordb.com/index.php?showid=1100000003593270725|http://factordb.com/cert.php?id=1100000003593270725|http://factordb.com/index.php?query=13*24%5E%28n%2B3%29%2B12013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3402|C7<sub>298</sub>|(283×24<sup>298</sup>−7)/23|299|413|http://factordb.com/index.php?id=1100000002326181235&open=prime|http://factordb.com/index.php?showid=1100000002326181235&base=24|http://factordb.com/index.php?showid=1100000002326181235|http://factordb.com/cert.php?id=1100000002326181235|http://factordb.com/index.php?query=%28283*24%5En-7%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3403|20<sub>313</sub>7|2×24<sup>314</sup>+7|315|434|http://factordb.com/index.php?id=1100000002355610241&open=prime|http://factordb.com/index.php?showid=1100000002355610241&base=24|http://factordb.com/index.php?showid=1100000002355610241|http://factordb.com/cert.php?id=1100000002355610241|http://factordb.com/index.php?query=2*24%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3404|BC0<sub>331</sub>B|276×24<sup>332</sup>+11|334|461|http://factordb.com/index.php?id=1100000002633359842&open=prime|http://factordb.com/index.php?showid=1100000002633359842&base=24|http://factordb.com/index.php?showid=1100000002633359842|http://factordb.com/cert.php?id=1100000002633359842|http://factordb.com/index.php?query=276*24%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3405|N<sub>2644</sub>LLN|24<sup>2647</sup>−1201|2647|3654|http://factordb.com/index.php?id=1100000003593270089&open=prime|http://factordb.com/index.php?showid=1100000003593270089&base=24|http://factordb.com/index.php?showid=1100000003593270089|http://factordb.com/cert.php?id=1100000003593270089|http://factordb.com/index.php?query=24%5E%28n%2B3%29-1201&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3406|D<sub>2698</sub>LD|(13×24<sup>2700</sup>+4403)/23|2700|3727|http://factordb.com/index.php?id=1100000003593269876&open=prime|http://factordb.com/index.php?showid=1100000003593269876&base=24|http://factordb.com/index.php?showid=1100000003593269876|http://factordb.com/cert.php?id=1100000003593269876|http://factordb.com/index.php?query=%2813*24%5E%28n%2B2%29%2B4403%29%2F23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3407|A0<sub>2951</sub>8ID|10×24<sup>2954</sup>+5053|2955|4079|http://factordb.com/index.php?id=1100000003593269654&open=prime|http://factordb.com/index.php?showid=1100000003593269654&base=24|http://factordb.com/index.php?showid=1100000003593269654|http://factordb.com/cert.php?id=1100000003593269654|http://factordb.com/index.php?query=10*24%5E%28n%2B3%29%2B5053&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3408|88N<sub>5951</sub>|201×24<sup>5951</sup>−1|5953|8216|http://factordb.com/index.php?id=1100000003593275880&open=prime|http://factordb.com/index.php?showid=1100000003593275880&base=24|http://factordb.com/index.php?showid=1100000003593275880|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=201*24%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|24|3409|N00N<sub>8129</sub>LN|13249×24<sup>8131</sup>−49|8134|11227|http://factordb.com/index.php?id=1100000003593391606&open=prime|http://factordb.com/index.php?showid=1100000003593391606&base=24|http://factordb.com/index.php?showid=1100000003593391606|http://factordb.com/cert.php?id=1100000003593391606|http://factordb.com/index.php?query=13249*24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25174|OL0<sub>214</sub>M9|645×26<sup>216</sup>+581|218|309|http://factordb.com/index.php?id=1100000000840631576&open=prime|http://factordb.com/index.php?showid=1100000000840631576&base=26|http://factordb.com/index.php?showid=1100000000840631576|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2<sup>2</sup>×5<sup>2</sup>×7×223×42849349×(296-digit prime)|http://factordb.com/index.php?query=645*26%5E%28n%2B2%29%2B581&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25175|1A<sub>219</sub>P|(7×26<sup>220</sup>+73)/5|221|312|http://factordb.com/index.php?id=1100000000840631595&open=prime|http://factordb.com/index.php?showid=1100000000840631595&base=26|http://factordb.com/index.php?showid=1100000000840631595|http://factordb.com/cert.php?id=1100000000840631595|http://factordb.com/index.php?query=%287*26%5E%28n%2B1%29%2B73%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25176|A<sub>223</sub>DP|(2×26<sup>225</sup>+463)/5|225|318|http://factordb.com/index.php?id=1100000003850155262&open=prime|http://factordb.com/index.php?showid=1100000003850155262&base=26|http://factordb.com/index.php?showid=1100000003850155262|http://factordb.com/cert.php?id=1100000003850155262|http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29%2B463%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25177|6J<sub>225</sub>|(169×26<sup>225</sup>−19)/25|226|320|http://factordb.com/index.php?id=1100000002328050895&open=prime|http://factordb.com/index.php?showid=1100000002328050895&base=26|http://factordb.com/index.php?showid=1100000002328050895|http://factordb.com/cert.php?id=1100000002328050895|http://factordb.com/index.php?query=%28169*26%5En-19%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25178|O<sub>228</sub>5|(24×26<sup>229</sup>−499)/25|229|325|http://factordb.com/index.php?id=1100000002328059255&open=prime|http://factordb.com/index.php?showid=1100000002328059255&base=26|http://factordb.com/index.php?showid=1100000002328059255|http://factordb.com/cert.php?id=1100000002328059255|http://factordb.com/index.php?query=%2824*26%5E%28n%2B1%29-499%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25179|K0<sub>230</sub>K0IP|20×26<sup>234</sup>+352013|235|333|http://factordb.com/index.php?id=1100000000840631669&open=prime|http://factordb.com/index.php?showid=1100000000840631669&base=26|http://factordb.com/index.php?showid=1100000000840631669|http://factordb.com/cert.php?id=1100000000840631669|http://factordb.com/index.php?query=20*26%5E%28n%2B4%29%2B352013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25180|B0<sub>236</sub>OB|11×26<sup>238</sup>+635|239|338|http://factordb.com/index.php?id=1100000002634136234&open=prime|http://factordb.com/index.php?showid=1100000002634136234&base=26|http://factordb.com/index.php?showid=1100000002634136234|http://factordb.com/cert.php?id=1100000002634136234|http://factordb.com/index.php?query=11*26%5E%28n%2B2%29%2B635&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25181|11G0<sub>239</sub>9|718×26<sup>240</sup>+9|243|343|http://factordb.com/index.php?id=1100000000840631687&open=prime|http://factordb.com/index.php?showid=1100000000840631687&base=26|http://factordb.com/index.php?showid=1100000000840631687|http://factordb.com/cert.php?id=1100000000840631687|http://factordb.com/index.php?query=718*26%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25182|K0<sub>241</sub>E5|20×26<sup>243</sup>+369|244|346|http://factordb.com/index.php?id=1100000002634136479&open=prime|http://factordb.com/index.php?showid=1100000002634136479&base=26|http://factordb.com/index.php?showid=1100000002634136479|http://factordb.com/cert.php?id=1100000002634136479|http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B369&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25183|J<sub>243</sub>0L|(19×26<sup>245</sup>−12319)/25|245|347|http://factordb.com/index.php?id=1100000003850155263&open=prime|http://factordb.com/index.php?showid=1100000003850155263&base=26|http://factordb.com/index.php?showid=1100000003850155263|http://factordb.com/cert.php?id=1100000003850155263|http://factordb.com/index.php?query=%2819*26%5E%28n%2B2%29-12319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25184|B<sub>251</sub>I9|(11×26<sup>253</sup>+4489)/25|253|358|http://factordb.com/index.php?id=1100000003850155264&open=prime|http://factordb.com/index.php?showid=1100000003850155264&base=26|http://factordb.com/index.php?showid=1100000003850155264|http://factordb.com/cert.php?id=1100000003850155264|http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B4489%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25185|F<sub>250</sub>0PCF|(3×26<sup>254</sup>−1284793)/5|254|360|http://factordb.com/index.php?id=1100000000840631708&open=prime|http://factordb.com/index.php?showid=1100000000840631708&base=26|http://factordb.com/index.php?showid=1100000000840631708|http://factordb.com/cert.php?id=1100000000840631708|http://factordb.com/index.php?query=%283*26%5E%28n%2B4%29-1284793%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25186|4E7<sub>262</sub>|(2957×26<sup>262</sup>−7)/25|264|373|http://factordb.com/index.php?id=1100000003850155265&open=prime|http://factordb.com/index.php?showid=1100000003850155265&base=26|http://factordb.com/index.php?showid=1100000003850155265|http://factordb.com/cert.php?id=1100000003850155265|http://factordb.com/index.php?query=%282957*26%5En-7%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25187|E7<sub>264</sub>OL|(357×26<sup>266</sup>+11393)/25|267|378|http://factordb.com/index.php?id=1100000003850155266&open=prime|http://factordb.com/index.php?showid=1100000003850155266&base=26|http://factordb.com/index.php?showid=1100000003850155266|http://factordb.com/cert.php?id=1100000003850155266|http://factordb.com/index.php?query=%28357*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25188|EIL<sub>267</sub>|(9571×26<sup>267</sup>−21)/25|269|381|http://factordb.com/index.php?id=1100000000840631801&open=prime|http://factordb.com/index.php?showid=1100000000840631801&base=26|http://factordb.com/index.php?showid=1100000000840631801|http://factordb.com/cert.php?id=1100000000840631801|http://factordb.com/index.php?query=%289571*26%5En-21%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25189|6<sub>268</sub>4F|(6×26<sup>270</sup>−1081)/25|270|382|http://factordb.com/index.php?id=1100000000840631976&open=prime|http://factordb.com/index.php?showid=1100000000840631976&base=26|http://factordb.com/index.php?showid=1100000000840631976|http://factordb.com/cert.php?id=1100000000840631976|http://factordb.com/index.php?query=%286*26%5E%28n%2B2%29-1081%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25190|D020<sub>273</sub>H|8790×26<sup>274</sup>+17|277|392|http://factordb.com/index.php?id=1100000003850155267&open=prime|http://factordb.com/index.php?showid=1100000003850155267&base=26|http://factordb.com/index.php?showid=1100000003850155267|http://factordb.com/cert.php?id=1100000003850155267|http://factordb.com/index.php?query=8790*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25191|B<sub>291</sub>KB|(11×26<sup>293</sup>+5839)/25|293|415|http://factordb.com/index.php?id=1100000003850155268&open=prime|http://factordb.com/index.php?showid=1100000003850155268&base=26|http://factordb.com/index.php?showid=1100000003850155268|http://factordb.com/cert.php?id=1100000003850155268|http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B5839%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25192|5<sub>293</sub>O5|(26<sup>295</sup>+2469)/5|295|417|http://factordb.com/index.php?id=1100000003850155269&open=prime|http://factordb.com/index.php?showid=1100000003850155269&base=26|http://factordb.com/index.php?showid=1100000003850155269|http://factordb.com/cert.php?id=1100000003850155269|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B2469%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25193|D7<sub>300</sub>|(332×26<sup>300</sup>−7)/25|301|426|http://factordb.com/index.php?id=1100000002328053362&open=prime|http://factordb.com/index.php?showid=1100000002328053362&base=26|http://factordb.com/index.php?showid=1100000002328053362|http://factordb.com/cert.php?id=1100000002328053362|http://factordb.com/index.php?query=%28332*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25194|E<sub>305</sub>IL|(14×26<sup>307</sup>+2761)/25|307|435|http://factordb.com/index.php?id=1100000000840632032&open=prime|http://factordb.com/index.php?showid=1100000000840632032&base=26|http://factordb.com/index.php?showid=1100000000840632032|http://factordb.com/cert.php?id=1100000000840632032|http://factordb.com/index.php?query=%2814*26%5E%28n%2B2%29%2B2761%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25195|PO5<sub>312</sub>|(3371×26<sup>312</sup>−1)/5|314|445|http://factordb.com/index.php?id=1100000003850155270&open=prime|http://factordb.com/index.php?showid=1100000003850155270&base=26|http://factordb.com/index.php?showid=1100000003850155270|http://factordb.com/cert.php?id=1100000003850155270|http://factordb.com/index.php?query=%283371*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25196|47<sub>314</sub>|(107×26<sup>314</sup>−7)/25|315|445|http://factordb.com/index.php?id=1100000002328050727&open=prime|http://factordb.com/index.php?showid=1100000002328050727&base=26|http://factordb.com/index.php?showid=1100000002328050727|http://factordb.com/cert.php?id=1100000002328050727|http://factordb.com/index.php?query=%28107*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25197|A<sub>335</sub>60F|(2×26<sup>338</sup>−14797)/5|338|478|http://factordb.com/index.php?id=1100000000840632163&open=prime|http://factordb.com/index.php?showid=1100000000840632163&base=26|http://factordb.com/index.php?showid=1100000000840632163|http://factordb.com/cert.php?id=1100000000840632163|http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-14797%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25198|O5K5<sub>341</sub>|(81871×26<sup>341</sup>−1)/5|344|487|http://factordb.com/index.php?id=1100000003850155271&open=prime|http://factordb.com/index.php?showid=1100000003850155271&base=26|http://factordb.com/index.php?showid=1100000003850155271|http://factordb.com/cert.php?id=1100000003850155271|http://factordb.com/index.php?query=%2887871*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25199|9K<sub>343</sub>AP|(49×26<sup>345</sup>−1279)/5|346|490|http://factordb.com/index.php?id=1100000000840632228&open=prime|http://factordb.com/index.php?showid=1100000000840632228&base=26|http://factordb.com/index.php?showid=1100000000840632228|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 1274/5×(26<sup>344</sup>−1), thus factor *N*−1 is equivalent to factor 26<sup>344</sup>−1, and for the factorization of 26<sup>344</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=|http://factordb.com/index.php?query=%2849*26%5E%28n%2B2%29-1279%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25200|8<sub>354</sub>1|(8×26<sup>355</sup>−183)/25|355|502|http://factordb.com/index.php?id=1100000000840632517&open=prime|http://factordb.com/index.php?showid=1100000000840632517&base=26|http://factordb.com/index.php?showid=1100000000840632517|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 208/25×(26<sup>354</sup>−1), thus factor *N*−1 is equivalent to factor 26<sup>354</sup>−1, and for the factorization of 26<sup>354</sup>−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=|http://factordb.com/index.php?query=%288*26%5E%28n%2B1%29-183%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25201|L0<sub>356</sub>66K9|21×26<sup>360</sup>+110041|361|511|http://factordb.com/index.php?id=1100000000840632748&open=prime|http://factordb.com/index.php?showid=1100000000840632748&base=26|http://factordb.com/index.php?showid=1100000000840632748|http://factordb.com/cert.php?id=1100000000840632748|http://factordb.com/index.php?query=21*26%5E%28n%2B4%29%2B110041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25202|K0<sub>358</sub>KIP|20×26<sup>361</sup>+14013|362|513|http://factordb.com/index.php?id=1100000000840632880&open=prime|http://factordb.com/index.php?showid=1100000000840632880&base=26|http://factordb.com/index.php?showid=1100000000840632880|http://factordb.com/cert.php?id=1100000000840632880|http://factordb.com/index.php?query=20*26%5E%28n%2B3%29%2B14013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25203|J0<sub>360</sub>A0P|19×26<sup>363</sup>+6785|364|515|http://factordb.com/index.php?id=1100000003850155272&open=prime|http://factordb.com/index.php?showid=1100000003850155272&base=26|http://factordb.com/index.php?showid=1100000003850155272|http://factordb.com/cert.php?id=1100000003850155272|http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B6785&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25204|OK6<sub>376</sub>9|(16106×26<sup>377</sup>+69)/25|379|537|http://factordb.com/index.php?id=1100000000840633320&open=prime|http://factordb.com/index.php?showid=1100000000840633320&base=26|http://factordb.com/index.php?showid=1100000000840633320|http://factordb.com/cert.php?id=1100000000840633320|http://factordb.com/index.php?query=%2816106*26%5E%28n%2B1%29%2B69%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25205|J<sub>406</sub>7|(19×26<sup>407</sup>−319)/25|407|576|http://factordb.com/index.php?id=1100000002328055467&open=prime|http://factordb.com/index.php?showid=1100000002328055467&base=26|http://factordb.com/index.php?showid=1100000002328055467|http://factordb.com/cert.php?id=1100000002328055467|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25206|9B<sub>437</sub>|(236×26<sup>437</sup>−11)/25|438|620|http://factordb.com/index.php?id=1100000002328051905&open=prime|http://factordb.com/index.php?showid=1100000002328051905&base=26|http://factordb.com/index.php?showid=1100000002328051905|http://factordb.com/cert.php?id=1100000002328051905|http://factordb.com/index.php?query=%28236*26%5En-11%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25207|3<sub>442</sub>GL|(3×26<sup>444</sup>+8897)/25|444|628|http://factordb.com/index.php?id=1100000003850155273&open=prime|http://factordb.com/index.php?showid=1100000003850155273&base=26|http://factordb.com/index.php?showid=1100000003850155273|http://factordb.com/cert.php?id=1100000003850155273|http://factordb.com/index.php?query=%283*26%5E%28n%2B2%29%2B8897%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25208|1M<sub>491</sub>P|(47×26<sup>492</sup>+53)/25|493|697|http://factordb.com/index.php?id=1100000000840633390&open=prime|http://factordb.com/index.php?showid=1100000000840633390&base=26|http://factordb.com/index.php?showid=1100000000840633390|http://factordb.com/cert.php?id=1100000000840633390|http://factordb.com/index.php?query=%2847*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25209|40<sub>509</sub>GL|4×26<sup>511</sup>+437|512|724|http://factordb.com/index.php?id=1100000000840633483&open=prime|http://factordb.com/index.php?showid=1100000000840633483&base=26|http://factordb.com/index.php?showid=1100000000840633483|http://factordb.com/cert.php?id=1100000000840633483|http://factordb.com/index.php?query=4*26%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25210|BFA<sub>511</sub>5|(1507×26<sup>512</sup>−27)/5|514|727|http://factordb.com/index.php?id=1100000003850155274&open=prime|http://factordb.com/index.php?showid=1100000003850155274&base=26|http://factordb.com/index.php?showid=1100000003850155274|http://factordb.com/cert.php?id=1100000003850155274|http://factordb.com/index.php?query=%281507*26%5E%28n%2B1%29-27%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25211|LK<sub>518</sub>5|(109×26<sup>519</sup>−79)/5|520|736|http://factordb.com/index.php?id=1100000003850155276&open=prime|http://factordb.com/index.php?showid=1100000003850155276&base=26|http://factordb.com/index.php?showid=1100000003850155276|http://factordb.com/cert.php?id=1100000003850155276|http://factordb.com/index.php?query=%28109*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25212|MI<sub>543</sub>3|(568×26<sup>544</sup>−393)/25|545|772|http://factordb.com/index.php?id=1100000003850155277&open=prime|http://factordb.com/index.php?showid=1100000003850155277&base=26|http://factordb.com/index.php?showid=1100000003850155277|http://factordb.com/cert.php?id=1100000003850155277|http://factordb.com/index.php?query=%28568*26%5E%28n%2B1%29-393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25213|E4E<sub>559</sub>7|(9214×26<sup>560</sup>−189)/25|562|795|http://factordb.com/index.php?id=1100000003850155278&open=prime|http://factordb.com/index.php?showid=1100000003850155278&base=26|http://factordb.com/index.php?showid=1100000003850155278|http://factordb.com/cert.php?id=1100000003850155278|http://factordb.com/index.php?query=%289214*26%5E%28n%2B1%29-189%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25214|80<sub>577</sub>C7|8×26<sup>579</sup>+319|580|821|http://factordb.com/index.php?id=1100000002634136160&open=prime|http://factordb.com/index.php?showid=1100000002634136160&base=26|http://factordb.com/index.php?showid=1100000002634136160|http://factordb.com/cert.php?id=1100000002634136160|http://factordb.com/index.php?query=8*26%5E%28n%2B2%29%2B319&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25215|9E0<sub>619</sub>B|248×26<sup>620</sup>+11|622|880|http://factordb.com/index.php?id=1100000002634136193&open=prime|http://factordb.com/index.php?showid=1100000002634136193&base=26|http://factordb.com/index.php?showid=1100000002634136193|http://factordb.com/cert.php?id=1100000002634136193|http://factordb.com/index.php?query=248*26%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25216|G60<sub>618</sub>KJ|422×26<sup>620</sup>+539|622|880|http://factordb.com/index.php?id=1100000003850155283&open=prime|http://factordb.com/index.php?showid=1100000003850155283&base=26|http://factordb.com/index.php?showid=1100000003850155283|http://factordb.com/cert.php?id=1100000003850155283|http://factordb.com/index.php?query=422*26%5E%28n%2B2%29%2B539&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25217|OO0<sub>620</sub>D3|648×26<sup>622</sup>+341|624|883|http://factordb.com/index.php?id=1100000003850155285&open=prime|http://factordb.com/index.php?showid=1100000003850155285&base=26|http://factordb.com/index.php?showid=1100000003850155285|http://factordb.com/cert.php?id=1100000003850155285|http://factordb.com/index.php?query=648*26%5E%28n%2B2%29%2B341&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25218|K0<sub>653</sub>IP|20×26<sup>655</sup>+493|656|929|http://factordb.com/index.php?id=1100000000840633594&open=prime|http://factordb.com/index.php?showid=1100000000840633594&base=26|http://factordb.com/index.php?showid=1100000000840633594|http://factordb.com/cert.php?id=1100000000840633594|http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B493&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25219|J00J<sub>698</sub>L|(321119×26<sup>699</sup>+31)/25|702|994|http://factordb.com/index.php?id=1100000003850155288&open=prime|http://factordb.com/index.php?showid=1100000003850155288&base=26|http://factordb.com/index.php?showid=1100000003850155288|http://factordb.com/cert.php?id=1100000003850155288|http://factordb.com/index.php?query=%28321119*26%5E%28n%2B1%29%2B31%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25220|B0<sub>772</sub>90J|11×26<sup>775</sup>+6103|776|1098|http://factordb.com/index.php?id=1100000003850155290&open=prime|http://factordb.com/index.php?showid=1100000003850155290&base=26|http://factordb.com/index.php?showid=1100000003850155290|http://factordb.com/cert.php?id=1100000003850155290|http://factordb.com/index.php?query=11*26%5E%28n%2B3%29%2B6103&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25221|J<sub>844</sub>B|(19×26<sup>845</sup>−219)/25|845|1196|http://factordb.com/index.php?id=1100000002328055693&open=prime|http://factordb.com/index.php?showid=1100000002328055693&base=26|http://factordb.com/index.php?showid=1100000002328055693|http://factordb.com/cert.php?id=1100000002328055693|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-219%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25222|H<sub>855</sub>M0H|(17×26<sup>858</sup>+73433)/25|858|1214|http://factordb.com/index.php?id=1100000003850155291&open=prime|http://factordb.com/index.php?showid=1100000003850155291&base=26|http://factordb.com/index.php?showid=1100000003850155291|http://factordb.com/cert.php?id=1100000003850155291|http://factordb.com/index.php?query=%2817*26%5E%28n%2B3%29%2B73433%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25223|J<sub>861</sub>OOL|(19×26<sup>864</sup>+87781)/25|864|1223|http://factordb.com/index.php?id=1100000003850155296&open=prime|http://factordb.com/index.php?showid=1100000003850155296&base=26|http://factordb.com/index.php?showid=1100000003850155296|http://factordb.com/cert.php?id=1100000003850155296|http://factordb.com/index.php?query=%2819*26%5E%28n%2B3%29%2B87781%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25224|B0<sub>979</sub>H|11×26<sup>980</sup>+17|981|1388|http://factordb.com/index.php?id=1100000002355639467&open=prime|http://factordb.com/index.php?showid=1100000002355639467&base=26|http://factordb.com/index.php?showid=1100000002355639467|http://factordb.com/cert.php?id=1100000002355639467|http://factordb.com/index.php?query=11*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25225|L0<sub>991</sub>4000J|21×26<sup>996</sup>+1827923|997|1411|http://factordb.com/index.php?id=1100000003850155301&open=prime|http://factordb.com/index.php?showid=1100000003850155301&base=26|http://factordb.com/index.php?showid=1100000003850155301|http://factordb.com/cert.php?id=1100000003850155301|http://factordb.com/index.php?query=21*26%5E%28n%2B5%29%2B1827923&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25226|E0<sub>1101</sub>K2B|14×26<sup>1104</sup>+13583|1105|1564|http://factordb.com/index.php?id=1100000003850155305&open=prime|http://factordb.com/index.php?showid=1100000003850155305&base=26|http://factordb.com/index.php?showid=1100000003850155305|http://factordb.com/cert.php?id=1100000003850155305|http://factordb.com/index.php?query=14*26%5E%28n%2B3%29%2B13583&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25227|G<sub>1105</sub>OO9|(16×26<sup>1108</sup>+140209)/25|1108|1568|http://factordb.com/index.php?id=1100000000840633717&open=prime|http://factordb.com/index.php?showid=1100000000840633717&base=26|http://factordb.com/index.php?showid=1100000000840633717|http://factordb.com/cert.php?id=1100000000840633717|http://factordb.com/index.php?query=%2816*26%5E%28n%2B3%29%2B140209%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25228|MC0<sub>1109</sub>N|584×26<sup>1110</sup>+23|1112|1574|http://factordb.com/index.php?id=1100000002634136576&open=prime|http://factordb.com/index.php?showid=1100000002634136576&base=26|http://factordb.com/index.php?showid=1100000002634136576|http://factordb.com/cert.php?id=1100000002634136576|http://factordb.com/index.php?query=584*26%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25229|G<sub>1159</sub>9|(16×26<sup>1160</sup>−191)/25|1160|1642|http://factordb.com/index.php?id=1100000000840633844&open=prime|http://factordb.com/index.php?showid=1100000000840633844&base=26|http://factordb.com/index.php?showid=1100000000840633844|http://factordb.com/cert.php?id=1100000000840633844|http://factordb.com/index.php?query=%2816*26%5E%28n%2B1%29-191%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25230|A<sub>1295</sub>06F|(2×26<sup>1298</sup>−34297)/5|1298|1837|http://factordb.com/index.php?id=1100000000840633998&open=prime|http://factordb.com/index.php?showid=1100000000840633998&base=26|http://factordb.com/index.php?showid=1100000000840633998|http://factordb.com/cert.php?id=1100000000840633998|http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-34297%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25231|KIA<sub>1298</sub>F|(2692×26<sup>1299</sup>+23)/5|1301|1841|http://factordb.com/index.php?id=1100000000840634108&open=prime|http://factordb.com/index.php?showid=1100000000840634108&base=26|http://factordb.com/index.php?showid=1100000000840634108|http://factordb.com/cert.php?id=1100000000840634108|http://factordb.com/index.php?query=%282692*26%5E%28n%2B1%29%2B23%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25232|L7<sub>1319</sub>OL|(532×26<sup>1321</sup>+11393)/25|1322|1871|http://factordb.com/index.php?id=1100000003850155311&open=prime|http://factordb.com/index.php?showid=1100000003850155311&base=26|http://factordb.com/index.php?showid=1100000003850155311|http://factordb.com/cert.php?id=1100000003850155311|http://factordb.com/index.php?query=%28532*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25233|J<sub>1365</sub>L|(19×26<sup>1366</sup>+31)/25|1366|1933|http://factordb.com/index.php?id=1100000002328055922&open=prime|http://factordb.com/index.php?showid=1100000002328055922&base=26|http://factordb.com/index.php?showid=1100000002328055922|http://factordb.com/cert.php?id=1100000002328055922|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B31%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25234|2<sub>1498</sub>H|(2×26<sup>1499</sup>+373)/25|1499|2120|http://factordb.com/index.php?id=1100000002328050300&open=prime|http://factordb.com/index.php?showid=1100000002328050300&base=26|http://factordb.com/index.php?showid=1100000002328050300|http://factordb.com/cert.php?id=1100000002328050300|http://factordb.com/index.php?query=%282*26%5E%28n%2B1%29%2B373%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25235|O5<sub>1509</sub>|(121×26<sup>1509</sup>−1)/5|1510|2137|http://factordb.com/index.php?id=1100000000894500022&open=prime|http://factordb.com/index.php?showid=1100000000894500022&base=26|http://factordb.com/index.php?showid=1100000000894500022|http://factordb.com/cert.php?id=1100000000894500022|http://factordb.com/index.php?query=%28121*26%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25236|DM<sub>1519</sub>P|(347×26<sup>1520</sup>+53)/25|1521|2152|http://factordb.com/index.php?id=1100000003850155312&open=prime|http://factordb.com/index.php?showid=1100000003850155312&base=26|http://factordb.com/index.php?showid=1100000003850155312|http://factordb.com/cert.php?id=1100000003850155312|http://factordb.com/index.php?query=%28347*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25237|J0<sub>1523</sub>P|19×26<sup>1524</sup>+25|1525|2158|http://factordb.com/index.php?id=1100000002355640604&open=prime|http://factordb.com/index.php?showid=1100000002355640604&base=26|http://factordb.com/index.php?showid=1100000002355640604|http://factordb.com/cert.php?id=1100000002355640604|http://factordb.com/index.php?query=19*26%5E%28n%2B1%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25238|F<sub>1569</sub>PCF|(3×26<sup>1572</sup>+33407)/5|1572|2225|http://factordb.com/index.php?id=1100000000840634210&open=prime|http://factordb.com/index.php?showid=1100000000840634210&base=26|http://factordb.com/index.php?showid=1100000000840634210|http://factordb.com/cert.php?id=1100000000840634210|http://factordb.com/index.php?query=%283*26%5E%28n%2B3%29%2B33407%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25239|N0<sub>1647</sub>NEN|23×26<sup>1650</sup>+15935|1651|2337|http://factordb.com/index.php?id=1100000003850155313&open=prime|http://factordb.com/index.php?showid=1100000003850155313&base=26|http://factordb.com/index.php?showid=1100000003850155313|http://factordb.com/cert.php?id=1100000003850155313|http://factordb.com/index.php?query=23*26%5E%28n%2B3%29%2B15935&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25240|5<sub>1885</sub>4P|(26<sup>1887</sup>−31)/5|1887|2670|http://factordb.com/index.php?id=1100000003850155314&open=prime|http://factordb.com/index.php?showid=1100000003850155314&base=26|http://factordb.com/index.php?showid=1100000003850155314|http://factordb.com/cert.php?id=1100000003850155314|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29-31%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25241|6K<sub>2556</sub>A5|(34×26<sup>2558</sup>−1379)/5|2559|3621|http://factordb.com/index.php?id=1100000003850155315&open=prime|http://factordb.com/index.php?showid=1100000003850155315&base=26|http://factordb.com/index.php?showid=1100000003850155315|http://factordb.com/cert.php?id=1100000003850155315|http://factordb.com/index.php?query=%2834*26%5E%28n%2B2%29-1379%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25242|70<sub>2613</sub>CN|7×26<sup>2615</sup>+335|2616|3702|http://factordb.com/index.php?id=1100000002634136105&open=prime|http://factordb.com/index.php?showid=1100000002634136105&base=26|http://factordb.com/index.php?showid=1100000002634136105|http://factordb.com/cert.php?id=1100000002634136105|http://factordb.com/index.php?query=7*26%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25243|E0<sub>2673</sub>H|14×26<sup>2674</sup>+17|2675|3785|http://factordb.com/index.php?id=1100000002355640062&open=prime|http://factordb.com/index.php?showid=1100000002355640062&base=26|http://factordb.com/index.php?showid=1100000002355640062|http://factordb.com/cert.php?id=1100000002355640062|http://factordb.com/index.php?query=14*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25244|G60<sub>2740</sub>J|422×26<sup>2741</sup>+19|2743|3882|http://factordb.com/index.php?id=1100000002634136363&open=prime|http://factordb.com/index.php?showid=1100000002634136363&base=26|http://factordb.com/index.php?showid=1100000002634136363|http://factordb.com/cert.php?id=1100000002634136363|http://factordb.com/index.php?query=422*26%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25245|B<sub>2915</sub>9|(11×26<sup>2916</sup>−61)/25|2916|4126|http://factordb.com/index.php?id=1100000002328052611&open=prime|http://factordb.com/index.php?showid=1100000002328052611&base=26|http://factordb.com/index.php?showid=1100000002328052611|http://factordb.com/cert.php?id=1100000002328052611|http://factordb.com/index.php?query=%2811*26%5E%28n%2B1%29-61%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25246|J<sub>4222</sub>P|(19×26<sup>4223</sup>+131)/25|4223|5976|http://factordb.com/index.php?id=1100000002328056865&open=prime|http://factordb.com/index.php?showid=1100000002328056865&base=26|http://factordb.com/index.php?showid=1100000002328056865|http://factordb.com/cert.php?id=1100000002328056865|http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B131%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25247|K0<sub>4364</sub>I5|20×26<sup>4366</sup>+473|4367|6180|http://factordb.com/index.php?id=1100000002634136508&open=prime|http://factordb.com/index.php?showid=1100000002634136508&base=26|http://factordb.com/index.php?showid=1100000002634136508|http://factordb.com/cert.php?id=1100000002634136508|http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25248|M<sub>8772</sub>P|(22×26<sup>8773</sup>+53)/25|8773|12414|http://factordb.com/index.php?id=1100000000758011195&open=prime|http://factordb.com/index.php?showid=1100000000758011195&base=26|http://factordb.com/index.php?showid=1100000000758011195|http://factordb.com/cert.php?id=1100000000758011195|http://factordb.com/index.php?query=%2822*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|26|25249|9GDK<sub>15920</sub>P|(32569×26<sup>15921</sup>+21)/5|15924|22532|http://factordb.com/index.php?id=1100000003850155316&open=prime|http://factordb.com/index.php?showid=1100000003850155316&base=26|http://factordb.com/index.php?showid=1100000003850155316|http://factordb.com/cert.php?id=1100000003850155316|http://factordb.com/index.php?query=%2832569*26%5E%28n%2B1%29%2B21%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25485|JN<sub>206</sub>|(536×28<sup>206</sup>−23)/27|207|300|http://factordb.com/index.php?id=1100000002611724435&open=prime|http://factordb.com/index.php?showid=1100000002611724435&base=28|http://factordb.com/index.php?showid=1100000002611724435|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2×1061×1171×74311×(289-digit prime)|http://factordb.com/index.php?query=%28536*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25486|3<sub>211</sub>M9|(28<sup>213</sup>+4841)/9|213|308|http://factordb.com/index.php?id=1100000003850161936&open=prime|http://factordb.com/index.php?showid=1100000003850161936&base=28|http://factordb.com/index.php?showid=1100000003850161936|http://factordb.com/cert.php?id=1100000003850161936|http://factordb.com/index.php?query=%2828%5E%28n%2B2%29%2B4841%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25487|HD0<sub>213</sub>D|489×28<sup>214</sup>+13|216|313|http://factordb.com/index.php?id=1100000003850161937&open=prime|http://factordb.com/index.php?showid=1100000003850161937&base=28|http://factordb.com/index.php?showid=1100000003850161937|http://factordb.com/cert.php?id=1100000003850161937|http://factordb.com/index.php?query=489*28%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25488|64O<sub>217</sub>9|(1556×28<sup>218</sup>−143)/9|220|318|http://factordb.com/index.php?id=1100000000840840215&open=prime|http://factordb.com/index.php?showid=1100000000840840215&base=28|http://factordb.com/index.php?showid=1100000000840840215|http://factordb.com/cert.php?id=1100000000840840215|http://factordb.com/index.php?query=%281556*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25489|G0<sub>217</sub>A0N|16×28<sup>220</sup>+7863|221|320|http://factordb.com/index.php?id=1100000003850161938&open=prime|http://factordb.com/index.php?showid=1100000003850161938&base=28|http://factordb.com/index.php?showid=1100000003850161938|http://factordb.com/cert.php?id=1100000003850161938|http://factordb.com/index.php?query=16*28%5E%28n%2B3%29%2B7863&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25490|55OA<sub>226</sub>F|(110278×28<sup>227</sup>+125)/27|230|333|http://factordb.com/index.php?id=1100000003850161939&open=prime|http://factordb.com/index.php?showid=1100000003850161939&base=28|http://factordb.com/index.php?showid=1100000003850161939|http://factordb.com/cert.php?id=1100000003850161939|http://factordb.com/index.php?query=%28110278*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25491|L0<sub>229</sub>Q3|21×28<sup>231</sup>+731|232|336|http://factordb.com/index.php?id=1100000003850161940&open=prime|http://factordb.com/index.php?showid=1100000003850161940&base=28|http://factordb.com/index.php?showid=1100000003850161940|http://factordb.com/cert.php?id=1100000003850161940|http://factordb.com/index.php?query=21*28%5E%28n%2B2%29%2B731&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25492|B0<sub>231</sub>7ID|11×28<sup>234</sup>+6005|235|340|http://factordb.com/index.php?id=1100000003850161941&open=prime|http://factordb.com/index.php?showid=1100000003850161941&base=28|http://factordb.com/index.php?showid=1100000003850161941|http://factordb.com/cert.php?id=1100000003850161941|http://factordb.com/index.php?query=11*28%5E%28n%2B3%29%2B6005&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25493|PM<sub>233</sub>B|(697×28<sup>234</sup>−319)/27|235|341|http://factordb.com/index.php?id=1100000003850161942&open=prime|http://factordb.com/index.php?showid=1100000003850161942&base=28|http://factordb.com/index.php?showid=1100000003850161942|http://factordb.com/cert.php?id=1100000003850161942|http://factordb.com/index.php?query=%28697*28%5E%28n%2B1%29-319%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25494|K0<sub>238</sub>OF|20×28<sup>240</sup>+687|241|349|http://factordb.com/index.php?id=1100000000840840142&open=prime|http://factordb.com/index.php?showid=1100000000840840142&base=28|http://factordb.com/index.php?showid=1100000000840840142|http://factordb.com/cert.php?id=1100000000840840142|http://factordb.com/index.php?query=20*28%5E%28n%2B2%29%2B687&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25495|I<sub>262</sub>E3|(2×28<sup>264</sup>−383)/3|264|382|http://factordb.com/index.php?id=1100000003850161943&open=prime|http://factordb.com/index.php?showid=1100000003850161943&base=28|http://factordb.com/index.php?showid=1100000003850161943|http://factordb.com/cert.php?id=1100000003850161943|http://factordb.com/index.php?query=%282*28%5E%28n%2B2%29-383%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25496|C5A<sub>273</sub>F|(9217×28<sup>274</sup>+125)/27|276|400|http://factordb.com/index.php?id=1100000003850161944&open=prime|http://factordb.com/index.php?showid=1100000003850161944&base=28|http://factordb.com/index.php?showid=1100000003850161944|http://factordb.com/cert.php?id=1100000003850161944|http://factordb.com/index.php?query=%289217*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25497|J0<sub>276</sub>IMB|19×28<sup>279</sup>+14739|280|406|http://factordb.com/index.php?id=1100000003850161945&open=prime|http://factordb.com/index.php?showid=1100000003850161945&base=28|http://factordb.com/index.php?showid=1100000003850161945|http://factordb.com/cert.php?id=1100000003850161945|http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B14739&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25498|F0<sub>282</sub>QAP|15×28<sup>285</sup>+20689|286|414|http://factordb.com/index.php?id=1100000000840840006&open=prime|http://factordb.com/index.php?showid=1100000000840840006&base=28|http://factordb.com/index.php?showid=1100000000840840006|http://factordb.com/cert.php?id=1100000000840840006|http://factordb.com/index.php?query=15*28%5E%28n%2B3%29%2B20689&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25499|M0<sub>296</sub>KKN|22×28<sup>299</sup>+16263|300|435|http://factordb.com/index.php?id=1100000003850161946&open=prime|http://factordb.com/index.php?showid=1100000003850161946&base=28|http://factordb.com/index.php?showid=1100000003850161946|http://factordb.com/cert.php?id=1100000003850161946|http://factordb.com/index.php?query=22*28%5E%28n%2B3%29%2B16263&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25500|C<sub>310</sub>43|(4×28<sup>312</sup>−2101)/9|312|452|http://factordb.com/index.php?id=1100000003850161947&open=prime|http://factordb.com/index.php?showid=1100000003850161947&base=28|http://factordb.com/index.php?showid=1100000003850161947|http://factordb.com/cert.php?id=1100000003850161947|http://factordb.com/index.php?query=%284*28%5E%28n%2B2%29-2101%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25501|RN<sub>319</sub>|(752×28<sup>319</sup>−23)/27|320|464|http://factordb.com/index.php?id=1100000002611723967&open=prime|http://factordb.com/index.php?showid=1100000002611723967&base=28|http://factordb.com/index.php?showid=1100000002611723967|http://factordb.com/cert.php?id=1100000002611723967|http://factordb.com/index.php?query=%28752*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25502|CA<sub>320</sub>F|(334×28<sup>321</sup>+125)/27|322|466|http://factordb.com/index.php?id=1100000000840839995&open=prime|http://factordb.com/index.php?showid=1100000000840839995&base=28|http://factordb.com/index.php?showid=1100000000840839995|http://factordb.com/cert.php?id=1100000000840839995|http://factordb.com/index.php?query=%28334*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25503|D6<sub>326</sub>LR|(119×28<sup>328</sup>+3967)/9|329|476|http://factordb.com/index.php?id=1100000003850161948&open=prime|http://factordb.com/index.php?showid=1100000003850161948&base=28|http://factordb.com/index.php?showid=1100000003850161948|http://factordb.com/cert.php?id=1100000003850161948|http://factordb.com/index.php?query=%28119*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25504|B<sub>350</sub>AB|(11×28<sup>352</sup>−767)/27|352|510|http://factordb.com/index.php?id=1100000003850161949&open=prime|http://factordb.com/index.php?showid=1100000003850161949&base=28|http://factordb.com/index.php?showid=1100000003850161949|http://factordb.com/cert.php?id=1100000003850161949|http://factordb.com/index.php?query=%2811*28%5E%28n%2B2%29-767%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25505|GA0<sub>355</sub>N|458×28<sup>356</sup>+23|358|518|http://factordb.com/index.php?id=1100000003850161950&open=prime|http://factordb.com/index.php?showid=1100000003850161950&base=28|http://factordb.com/index.php?showid=1100000003850161950|http://factordb.com/cert.php?id=1100000003850161950|http://factordb.com/index.php?query=458*28%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25506|A0<sub>356</sub>P7P|10×28<sup>359</sup>+19821|360|521|http://factordb.com/index.php?id=1100000003850161951&open=prime|http://factordb.com/index.php?showid=1100000003850161951&base=28|http://factordb.com/index.php?showid=1100000003850161951|http://factordb.com/cert.php?id=1100000003850161951|http://factordb.com/index.php?query=10*28%5E%28n%2B3%29%2B19821&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25507|J<sub>363</sub>H|(19×28<sup>364</sup>−73)/27|364|527|http://factordb.com/index.php?id=1100000002611724460&open=prime|http://factordb.com/index.php?showid=1100000002611724460&base=28|http://factordb.com/index.php?showid=1100000002611724460|http://factordb.com/cert.php?id=1100000002611724460|http://factordb.com/index.php?query=%2819*28%5E%28n%2B1%29-73%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25508|4B<sub>381</sub>|(119×28<sup>381</sup>−11)/27|382|553|http://factordb.com/index.php?id=1100000002611724588&open=prime|http://factordb.com/index.php?showid=1100000002611724588&base=28|http://factordb.com/index.php?showid=1100000002611724588|http://factordb.com/cert.php?id=1100000002611724588|http://factordb.com/index.php?query=%28119*28%5En-11%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25509|EB0<sub>405</sub>1|403×28<sup>406</sup>+1|408|591|http://factordb.com/index.php?id=1100000001534442374&open=prime|http://factordb.com/index.php?showid=1100000001534442374&base=28|http://factordb.com/index.php?showid=1100000001534442374|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=403*28%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25510|AN<sub>461</sub>|(293×28<sup>461</sup>−23)/27|462|669|http://factordb.com/index.php?id=1100000002611724556&open=prime|http://factordb.com/index.php?showid=1100000002611724556&base=28|http://factordb.com/index.php?showid=1100000002611724556|http://factordb.com/cert.php?id=1100000002611724556|http://factordb.com/index.php?query=%28293*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25511|4O<sub>614</sub>09|(44×28<sup>616</sup>−6191)/9|617|893|http://factordb.com/index.php?id=1100000000840839989&open=prime|http://factordb.com/index.php?showid=1100000000840839989&base=28|http://factordb.com/index.php?showid=1100000000840839989|http://factordb.com/cert.php?id=1100000000840839989|http://factordb.com/index.php?query=%2844*28%5E%28n%2B2%29-6191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25512|2D<sub>641</sub>|(67×28<sup>641</sup>−13)/27|642|929|http://factordb.com/index.php?id=1100000002611725341&open=prime|http://factordb.com/index.php?showid=1100000002611725341&base=28|http://factordb.com/index.php?showid=1100000002611725341|http://factordb.com/cert.php?id=1100000002611725341|http://factordb.com/index.php?query=%2867*28%5En-13%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25513|70<sub>748</sub>M5|7×28<sup>750</sup>+621|751|1087|http://factordb.com/index.php?id=1100000003850161956&open=prime|http://factordb.com/index.php?showid=1100000003850161956&base=28|http://factordb.com/index.php?showid=1100000003850161956|http://factordb.com/cert.php?id=1100000003850161956|http://factordb.com/index.php?query=7*28%5E%28n%2B2%29%2B621&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25514|4A0<sub>804</sub>B|122×28<sup>805</sup>+11|807|1168|http://factordb.com/index.php?id=1100000003850161957&open=prime|http://factordb.com/index.php?showid=1100000003850161957&base=28|http://factordb.com/index.php?showid=1100000003850161957|http://factordb.com/cert.php?id=1100000003850161957|http://factordb.com/index.php?query=122*28%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25515|LK<sub>925</sub>F|(587×28<sup>926</sup>−155)/27|927|1342|http://factordb.com/index.php?id=1100000000840839978&open=prime|http://factordb.com/index.php?showid=1100000000840839978&base=28|http://factordb.com/index.php?showid=1100000000840839978|http://factordb.com/cert.php?id=1100000000840839978|http://factordb.com/index.php?query=%28587*28%5E%28n%2B1%29-155%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25516|J0<sub>1071</sub>AC5|19×28<sup>1074</sup>+8181|1075|1556|http://factordb.com/index.php?id=1100000003850161959&open=prime|http://factordb.com/index.php?showid=1100000003850161959&base=28|http://factordb.com/index.php?showid=1100000003850161959|http://factordb.com/cert.php?id=1100000003850161959|http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B8181&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25517|J0<sub>1252</sub>J5|19×28<sup>1254</sup>+537|1255|1817|http://factordb.com/index.php?id=1100000003850161963&open=prime|http://factordb.com/index.php?showid=1100000003850161963&base=28|http://factordb.com/index.php?showid=1100000003850161963|http://factordb.com/cert.php?id=1100000003850161963|http://factordb.com/index.php?query=19*28%5E%28n%2B2%29%2B537&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25518|5<sub>1304</sub>6F|(5×28<sup>1306</sup>+1021)/27|1306|1890|http://factordb.com/index.php?id=1100000003850161964&open=prime|http://factordb.com/index.php?showid=1100000003850161964&base=28|http://factordb.com/index.php?showid=1100000003850161964|http://factordb.com/cert.php?id=1100000003850161964|http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B1021%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25519|5<sub>1332</sub>P8P|(5×28<sup>1335</sup>+426163)/27|1335|1932|http://factordb.com/index.php?id=1100000003850161965&open=prime|http://factordb.com/index.php?showid=1100000003850161965&base=28|http://factordb.com/index.php?showid=1100000003850161965|http://factordb.com/cert.php?id=1100000003850161965|http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25520|5I<sub>1370</sub>F|(17×28<sup>1371</sup>−11)/3|1372|1985|http://factordb.com/index.php?id=1100000003850161972&open=prime|http://factordb.com/index.php?showid=1100000003850161972&base=28|http://factordb.com/index.php?showid=1100000003850161972|http://factordb.com/cert.php?id=1100000003850161972|http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25521|A<sub>1423</sub>6F|(10×28<sup>1425</sup>−2899)/27|1425|2062|http://factordb.com/index.php?id=1100000000840839947&open=prime|http://factordb.com/index.php?showid=1100000000840839947&base=28|http://factordb.com/index.php?showid=1100000000840839947|http://factordb.com/cert.php?id=1100000000840839947|http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25522|G0<sub>1899</sub>AN|16×28<sup>1901</sup>+303|1902|2753|http://factordb.com/index.php?id=1100000003850161973&open=prime|http://factordb.com/index.php?showid=1100000003850161973&base=28|http://factordb.com/index.php?showid=1100000003850161973|http://factordb.com/cert.php?id=1100000003850161973|http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25523|5<sub>3746</sub>8P|(5×28<sup>3748</sup>+2803)/27|3748|5424|http://factordb.com/index.php?id=1100000003850161974&open=prime|http://factordb.com/index.php?showid=1100000003850161974&base=28|http://factordb.com/index.php?showid=1100000003850161974|http://factordb.com/cert.php?id=1100000003850161974|http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25524|QO<sub>4239</sub>69|(242×28<sup>4241</sup>−4679)/9|4242|6139|http://factordb.com/index.php?id=1100000000840839934&open=prime|http://factordb.com/index.php?showid=1100000000840839934&base=28|http://factordb.com/index.php?showid=1100000000840839934|http://factordb.com/cert.php?id=1100000000840839934|http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|28|25525|D0<sub>5267</sub>77D|13×28<sup>5270</sup>+5697|5271|7628|http://factordb.com/index.php?id=1100000003850151420&open=prime|http://factordb.com/index.php?showid=1100000003850151420&base=28|http://factordb.com/index.php?showid=1100000003850151420|http://factordb.com/cert.php?id=1100000003850151420|http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2613|AN<sub>206</sub>|(313×30<sup>206</sup>−23)/29|207|306|http://factordb.com/index.php?id=1100000002327651073&open=prime|http://factordb.com/index.php?showid=1100000002327651073&base=30|http://factordb.com/index.php?showid=1100000002327651073|http://factordb.com/cert.php?id=1100000002327651073|http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2614|M<sub>241</sub>QB|(22×30<sup>243</sup>+3139)/29|243|359|http://factordb.com/index.php?id=1100000003593408295&open=prime|http://factordb.com/index.php?showid=1100000003593408295&base=30|http://factordb.com/index.php?showid=1100000003593408295|http://factordb.com/cert.php?id=1100000003593408295|http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2615|M0<sub>547</sub>SS7|22×30<sup>550</sup>+26047|551|814|http://factordb.com/index.php?id=1100000003593407988&open=prime|http://factordb.com/index.php?showid=1100000003593407988&base=30|http://factordb.com/index.php?showid=1100000003593407988|http://factordb.com/cert.php?id=1100000003593407988|http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2616|C0<sub>1022</sub>1|12×30<sup>1023</sup>+1|1024|1513|http://factordb.com/index.php?id=1100000000785448736&open=prime|http://factordb.com/index.php?showid=1100000000785448736&base=30|http://factordb.com/index.php?showid=1100000000785448736|proven prime by *N*−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=12*30%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2617|5<sub>4882</sub>J|(5×30<sup>4883</sup>+401)/29|4883|7213|http://factordb.com/index.php?id=1100000002327649423&open=prime|http://factordb.com/index.php?showid=1100000002327649423&base=30|http://factordb.com/index.php?showid=1100000002327649423|http://factordb.com/cert.php?id=1100000002327649423|http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|30|2619|OT<sub>34205</sub>|25×30<sup>34205</sup>−1|34206|50527|http://factordb.com/index.php?id=1100000000800812865&open=prime|http://factordb.com/index.php?showid=1100000000800812865&base=30|http://factordb.com/index.php?showid=1100000000800812865|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35212|Q<sub>195</sub>77|(26×36<sup>197</sup>−24631)/35|197|307|http://factordb.com/index.php?id=1100000003807362350&open=prime|http://factordb.com/index.php?showid=1100000003807362350&base=36|http://factordb.com/index.php?showid=1100000003807362350|http://factordb.com/cert.php?id=1100000003807362350|http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-24631%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35213|W0<sub>199</sub>ND|32×36<sup>201</sup>+841|202|315|http://factordb.com/index.php?id=1100000002634136732&open=prime|http://factordb.com/index.php?showid=1100000002634136732&base=36|http://factordb.com/index.php?showid=1100000002634136732|http://factordb.com/cert.php?id=1100000002634136732|http://factordb.com/index.php?query=32*36%5E%28n%2B2%29%2B841&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35214|G0<sub>204</sub>YT|16×36<sup>206</sup>+1253|207|322|http://factordb.com/index.php?id=1100000002634137789&open=prime|http://factordb.com/index.php?showid=1100000002634137789&base=36|http://factordb.com/index.php?showid=1100000002634137789|http://factordb.com/cert.php?id=1100000002634137789|http://factordb.com/index.php?query=16*36%5E%28n%2B2%29%2B1253&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35215|RHY<sub>223</sub>H|(34649×36<sup>224</sup>−629)/35|226|352|http://factordb.com/index.php?id=1100000003807362353&open=prime|http://factordb.com/index.php?showid=1100000003807362353&base=36|http://factordb.com/index.php?showid=1100000003807362353|http://factordb.com/cert.php?id=1100000003807362353|http://factordb.com/index.php?query=%2834649*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35216|T0<sub>223</sub>ST|29×36<sup>225</sup>+1037|226|352|http://factordb.com/index.php?id=1100000002634136882&open=prime|http://factordb.com/index.php?showid=1100000002634136882&base=36|http://factordb.com/index.php?showid=1100000002634136882|http://factordb.com/cert.php?id=1100000002634136882|http://factordb.com/index.php?query=29*36%5E%28n%2B2%29%2B1037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35217|J0<sub>224</sub>U6V|19×36<sup>227</sup>+39127|228|355|http://factordb.com/index.php?id=1100000003807362355&open=prime|http://factordb.com/index.php?showid=1100000003807362355&base=36|http://factordb.com/index.php?showid=1100000003807362355|http://factordb.com/cert.php?id=1100000003807362355|http://factordb.com/index.php?query=19*36%5E%28n%2B3%29%2B39127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35218|BE0<sub>235</sub>IV|410×36<sup>237</sup>+679|239|372|http://factordb.com/index.php?id=1100000003807362356&open=prime|http://factordb.com/index.php?showid=1100000003807362356&base=36|http://factordb.com/index.php?showid=1100000003807362356|http://factordb.com/cert.php?id=1100000003807362356|http://factordb.com/index.php?query=410*36%5E%28n%2B2%29%2B679&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35219|E0<sub>236</sub>KY1|14×36<sup>239</sup>+27145|240|374|http://factordb.com/index.php?id=1100000000840634520&open=prime|http://factordb.com/index.php?showid=1100000000840634520&base=36|http://factordb.com/index.php?showid=1100000000840634520|http://factordb.com/cert.php?id=1100000000840634520|http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B27145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35220|JXV0<sub>244</sub>B|25843×36<sup>245</sup>+11|248|386|http://factordb.com/index.php?id=1100000003807362357&open=prime|http://factordb.com/index.php?showid=1100000003807362357&base=36|http://factordb.com/index.php?showid=1100000003807362357|http://factordb.com/cert.php?id=1100000003807362357|http://factordb.com/index.php?query=25843*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35221|5Q<sub>249</sub>7|(201×36<sup>250</sup>−691)/35|251|390|http://factordb.com/index.php?id=1100000003807362359&open=prime|http://factordb.com/index.php?showid=1100000003807362359&base=36|http://factordb.com/index.php?showid=1100000003807362359|http://factordb.com/cert.php?id=1100000003807362359|http://factordb.com/index.php?query=%28201*36%5E%28n%2B1%29-691%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35222|N30<sub>253</sub>H|831×36<sup>254</sup>+17|256|399|http://factordb.com/index.php?id=1100000002634137359&open=prime|http://factordb.com/index.php?showid=1100000002634137359&base=36|http://factordb.com/index.php?showid=1100000002634137359|http://factordb.com/cert.php?id=1100000002634137359|http://factordb.com/index.php?query=831*36%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35223|Y<sub>261</sub>AH|(34×36<sup>263</sup>−30869)/35|263|410|http://factordb.com/index.php?id=1100000003807362360&open=prime|http://factordb.com/index.php?showid=1100000003807362360&base=36|http://factordb.com/index.php?showid=1100000003807362360|http://factordb.com/cert.php?id=1100000003807362360|http://factordb.com/index.php?query=%2834*36%5E%28n%2B2%29-30869%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35224|90<sub>277</sub>VV|9×36<sup>279</sup>+1147|280|436|http://factordb.com/index.php?id=1100000002634138388&open=prime|http://factordb.com/index.php?showid=1100000002634138388&base=36|http://factordb.com/index.php?showid=1100000002634138388|http://factordb.com/cert.php?id=1100000002634138388|http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B1147&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35225|J0<sub>281</sub>VB|19×36<sup>283</sup>+1127|284|442|http://factordb.com/index.php?id=1100000002634137683&open=prime|http://factordb.com/index.php?showid=1100000002634137683&base=36|http://factordb.com/index.php?showid=1100000002634137683|http://factordb.com/cert.php?id=1100000002634137683|http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35226|J0<sub>281</sub>WV|19×36<sup>283</sup>+1183|284|442|http://factordb.com/index.php?id=1100000002634137660&open=prime|http://factordb.com/index.php?showid=1100000002634137660&base=36|http://factordb.com/index.php?showid=1100000002634137660|http://factordb.com/cert.php?id=1100000002634137660|http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35227|DE0<sub>281</sub>61|482×36<sup>283</sup>+217|285|444|http://factordb.com/index.php?id=1100000003807362361&open=prime|http://factordb.com/index.php?showid=1100000003807362361&base=36|http://factordb.com/index.php?showid=1100000003807362361|http://factordb.com/cert.php?id=1100000003807362361|http://factordb.com/index.php?query=482*36%5E%28n%2B2%29%2B217&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35228|9H<sub>297</sub>|(332×36<sup>297</sup>−17)/35|298|464|http://factordb.com/index.php?id=1100000002332535884&open=prime|http://factordb.com/index.php?showid=1100000002332535884&base=36|http://factordb.com/index.php?showid=1100000002332535884|http://factordb.com/cert.php?id=1100000002332535884|http://factordb.com/index.php?query=%28332*36%5En-17%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35229|M70<sub>297</sub>FD|799×36<sup>299</sup>+553|301|469|http://factordb.com/index.php?id=1100000003807362363&open=prime|http://factordb.com/index.php?showid=1100000003807362363&base=36|http://factordb.com/index.php?showid=1100000003807362363|http://factordb.com/cert.php?id=1100000003807362363|http://factordb.com/index.php?query=799*36%5E%28n%2B2%29%2B553&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35230|9X<sub>301</sub>B|(348×36<sup>302</sup>−803)/35|303|472|http://factordb.com/index.php?id=1100000003807362364&open=prime|http://factordb.com/index.php?showid=1100000003807362364&base=36|http://factordb.com/index.php?showid=1100000003807362364|http://factordb.com/cert.php?id=1100000003807362364|http://factordb.com/index.php?query=%28348*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35231|XE0<sub>325</sub>7|1202×36<sup>326</sup>+7|328|511|http://factordb.com/index.php?id=1100000002634136674&open=prime|http://factordb.com/index.php?showid=1100000002634136674&base=36|http://factordb.com/index.php?showid=1100000002634136674|http://factordb.com/cert.php?id=1100000002634136674|http://factordb.com/index.php?query=1202*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35232|KP<sub>330</sub>SZ|(145×36<sup>332</sup>+821)/7|333|519|http://factordb.com/index.php?id=1100000000840634515&open=prime|http://factordb.com/index.php?showid=1100000000840634515&base=36|http://factordb.com/index.php?showid=1100000000840634515|http://factordb.com/cert.php?id=1100000000840634515|http://factordb.com/index.php?query=%28145*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35233|5<sub>347</sub>QP|(36<sup>349</sup>+5431)/7|349|543|http://factordb.com/index.php?id=1100000003807362365&open=prime|http://factordb.com/index.php?showid=1100000003807362365&base=36|http://factordb.com/index.php?showid=1100000003807362365|http://factordb.com/cert.php?id=1100000003807362365|http://factordb.com/index.php?query=%2836%5E%28n%2B2%29%2B5431%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35234|E0<sub>349</sub>6U1|14×36<sup>352</sup>+8857|353|549|http://factordb.com/index.php?id=1100000000840634509&open=prime|http://factordb.com/index.php?showid=1100000000840634509&base=36|http://factordb.com/index.php?showid=1100000000840634509|http://factordb.com/cert.php?id=1100000000840634509|http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B8857&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35235|K0<sub>367</sub>E6T|20×36<sup>370</sup>+18389|371|578|http://factordb.com/index.php?id=1100000003807362367&open=prime|http://factordb.com/index.php?showid=1100000003807362367&base=36|http://factordb.com/index.php?showid=1100000003807362367|http://factordb.com/cert.php?id=1100000003807362367|http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B18389&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35236|U0<sub>370</sub>WP|30×36<sup>372</sup>+1177|373|581|http://factordb.com/index.php?id=1100000000840634503&open=prime|http://factordb.com/index.php?showid=1100000000840634503&base=36|http://factordb.com/index.php?showid=1100000000840634503|http://factordb.com/cert.php?id=1100000000840634503|http://factordb.com/index.php?query=30*36%5E%28n%2B2%29%2B1177&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35237|P8Z<sub>390</sub>|909×36<sup>390</sup>−1|392|610|http://factordb.com/index.php?id=1100000000764100228&open=prime|http://factordb.com/index.php?showid=1100000000764100228&base=36|http://factordb.com/index.php?showid=1100000000764100228|proven prime by *N*+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=909*36%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35238|90<sub>397</sub>4B|9×36<sup>399</sup>+155|400|622|http://factordb.com/index.php?id=1100000002634138490&open=prime|http://factordb.com/index.php?showid=1100000002634138490&base=36|http://factordb.com/index.php?showid=1100000002634138490|http://factordb.com/cert.php?id=1100000002634138490|http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B155&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35239|50<sub>405</sub>WW5|5×36<sup>408</sup>+42629|409|636|http://factordb.com/index.php?id=1100000003807362369&open=prime|http://factordb.com/index.php?showid=1100000003807362369&base=36|http://factordb.com/index.php?showid=1100000003807362369|http://factordb.com/cert.php?id=1100000003807362369|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42629&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35240|V0<sub>405</sub>EE4B|31×36<sup>409</sup>+671483|410|639|http://factordb.com/index.php?id=1100000003807362370&open=prime|http://factordb.com/index.php?showid=1100000003807362370&base=36|http://factordb.com/index.php?showid=1100000003807362370|http://factordb.com/cert.php?id=1100000003807362370|http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B671483&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35241|TTR0<sub>434</sub>T|38655×36<sup>435</sup>+29|438|682|http://factordb.com/index.php?id=1100000003807362372&open=prime|http://factordb.com/index.php?showid=1100000003807362372&base=36|http://factordb.com/index.php?showid=1100000003807362372|http://factordb.com/cert.php?id=1100000003807362372|http://factordb.com/index.php?query=38655*36%5E%28n%2B1%29%2B29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35242|LK<sub>443</sub>Z|(151×36<sup>444</sup>+101)/7|445|693|http://factordb.com/index.php?id=1100000000840634496&open=prime|http://factordb.com/index.php?showid=1100000000840634496&base=36|http://factordb.com/index.php?showid=1100000000840634496|http://factordb.com/cert.php?id=1100000000840634496|http://factordb.com/index.php?query=%28151*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35243|Q<sub>454</sub>7|(26×36<sup>455</sup>−691)/35|455|708|http://factordb.com/index.php?id=1100000002332534290&open=prime|http://factordb.com/index.php?showid=1100000002332534290&base=36|http://factordb.com/index.php?showid=1100000002332534290|http://factordb.com/cert.php?id=1100000002332534290|http://factordb.com/index.php?query=%2826*36%5E%28n%2B1%29-691%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35244|R0<sub>458</sub>7|27×36<sup>459</sup>+7|460|716|http://factordb.com/index.php?id=1100000002356257765&open=prime|http://factordb.com/index.php?showid=1100000002356257765&base=36|http://factordb.com/index.php?showid=1100000002356257765|http://factordb.com/cert.php?id=1100000002356257765|http://factordb.com/index.php?query=27*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35245|50<sub>460</sub>KGZ|5×36<sup>463</sup>+26531|464|722|http://factordb.com/index.php?id=1100000003807362374&open=prime|http://factordb.com/index.php?showid=1100000003807362374&base=36|http://factordb.com/index.php?showid=1100000003807362374|http://factordb.com/cert.php?id=1100000003807362374|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B26531&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35246|K0<sub>474</sub>OY1|20×36<sup>477</sup>+32329|478|744|http://factordb.com/index.php?id=1100000000840634488&open=prime|http://factordb.com/index.php?showid=1100000000840634488&base=36|http://factordb.com/index.php?showid=1100000000840634488|http://factordb.com/cert.php?id=1100000000840634488|http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B32329&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35247|B<sub>478</sub>KB|(11×36<sup>480</sup>+11329)/35|480|747|http://factordb.com/index.php?id=1100000003807362381&open=prime|http://factordb.com/index.php?showid=1100000003807362381&base=36|http://factordb.com/index.php?showid=1100000003807362381|http://factordb.com/cert.php?id=1100000003807362381|http://factordb.com/index.php?query=%2811*36%5E%28n%2B2%29%2B11329%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35248|WY<sub>507</sub>H|(1154×36<sup>508</sup>−629)/35|509|793|http://factordb.com/index.php?id=1100000003807362386&open=prime|http://factordb.com/index.php?showid=1100000003807362386&base=36|http://factordb.com/index.php?showid=1100000003807362386|http://factordb.com/cert.php?id=1100000003807362386|http://factordb.com/index.php?query=%281154*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35249|G0<sub>510</sub>USJ|16×36<sup>513</sup>+39907|514|800|http://factordb.com/index.php?id=1100000003807362389&open=prime|http://factordb.com/index.php?showid=1100000003807362389&base=36|http://factordb.com/index.php?showid=1100000003807362389|http://factordb.com/cert.php?id=1100000003807362389|http://factordb.com/index.php?query=16*36%5E%28n%2B3%29%2B39907&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35250|Z<sub>527</sub>EX7|36<sup>530</sup>−27317|530|825|http://factordb.com/index.php?id=1100000003807362391&open=prime|http://factordb.com/index.php?showid=1100000003807362391&base=36|http://factordb.com/index.php?showid=1100000003807362391|http://factordb.com/cert.php?id=1100000003807362391|http://factordb.com/index.php?query=36%5E%28n%2B3%29-27317&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35251|EY0<sub>534</sub>A1|538×36<sup>536</sup>+361|538|837|http://factordb.com/index.php?id=1100000000840634482&open=prime|http://factordb.com/index.php?showid=1100000000840634482&base=36|http://factordb.com/index.php?showid=1100000000840634482|http://factordb.com/cert.php?id=1100000000840634482|http://factordb.com/index.php?query=538*36%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35252|Z0<sub>563</sub>995|35×36<sup>566</sup>+11993|567|883|http://factordb.com/index.php?id=1100000003807362394&open=prime|http://factordb.com/index.php?showid=1100000003807362394&base=36|http://factordb.com/index.php?showid=1100000003807362394|http://factordb.com/cert.php?id=1100000003807362394|http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B11993&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35253|F<sub>590</sub>95|(3×36<sup>592</sup>−1585)/7|592|921|http://factordb.com/index.php?id=1100000003807362398&open=prime|http://factordb.com/index.php?showid=1100000003807362398&base=36|http://factordb.com/index.php?showid=1100000003807362398|http://factordb.com/cert.php?id=1100000003807362398|http://factordb.com/index.php?query=%283*36%5E%28n%2B2%29-1585%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35254|990<sub>591</sub>B|333×36<sup>592</sup>+11|594|924|http://factordb.com/index.php?id=1100000002634138415&open=prime|http://factordb.com/index.php?showid=1100000002634138415&base=36|http://factordb.com/index.php?showid=1100000002634138415|http://factordb.com/cert.php?id=1100000002634138415|http://factordb.com/index.php?query=333*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35255|J<sub>675</sub>T|(19×36<sup>676</sup>+331)/35|676|1052|http://factordb.com/index.php?id=1100000002332534943&open=prime|http://factordb.com/index.php?showid=1100000002332534943&base=36|http://factordb.com/index.php?showid=1100000002332534943|http://factordb.com/cert.php?id=1100000002332534943|http://factordb.com/index.php?query=%2819*36%5E%28n%2B1%29%2B331%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35256|FZ<sub>708</sub>OEB|16×36<sup>711</sup>−15037|712|1108|http://factordb.com/index.php?id=1100000003807362403&open=prime|http://factordb.com/index.php?showid=1100000003807362403&base=36|http://factordb.com/index.php?showid=1100000003807362403|http://factordb.com/cert.php?id=1100000003807362403|http://factordb.com/index.php?query=16*36%5E%28n%2B3%29-15037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35257|EX<sub>732</sub>B|(523×36<sup>733</sup>−803)/35|734|1142|http://factordb.com/index.php?id=1100000003807362408&open=prime|http://factordb.com/index.php?showid=1100000003807362408&base=36|http://factordb.com/index.php?showid=1100000003807362408|http://factordb.com/cert.php?id=1100000003807362408|http://factordb.com/index.php?query=%28523*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35258|M<sub>764</sub>8B|(22×36<sup>766</sup>−18047)/35|766|1192|http://factordb.com/index.php?id=1100000003807362414&open=prime|http://factordb.com/index.php?showid=1100000003807362414&base=36|http://factordb.com/index.php?showid=1100000003807362414|http://factordb.com/cert.php?id=1100000003807362414|http://factordb.com/index.php?query=%2822*36%5E%28n%2B2%29-18047%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35259|3EG<sub>777</sub>D|(4286×36<sup>778</sup>−121)/35|780|1213|http://factordb.com/index.php?id=1100000003807362419&open=prime|http://factordb.com/index.php?showid=1100000003807362419&base=36|http://factordb.com/index.php?showid=1100000003807362419|http://factordb.com/cert.php?id=1100000003807362419|http://factordb.com/index.php?query=%284286*36%5E%28n%2B1%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35260|W<sub>962</sub>7|(32×36<sup>963</sup>−907)/35|963|1499|http://factordb.com/index.php?id=1100000002332533447&open=prime|http://factordb.com/index.php?showid=1100000002332533447&base=36|http://factordb.com/index.php?showid=1100000002332533447|http://factordb.com/cert.php?id=1100000002332533447|http://factordb.com/index.php?query=%2832*36%5E%28n%2B1%29-907%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35261|G<sub>979</sub>88D|(16×36<sup>982</sup>−373081)/35|982|1528|http://factordb.com/index.php?id=1100000003807362435&open=prime|http://factordb.com/index.php?showid=1100000003807362435&base=36|http://factordb.com/index.php?showid=1100000003807362435|http://factordb.com/cert.php?id=1100000003807362435|http://factordb.com/index.php?query=%2816*36%5E%28n%2B3%29-373081%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35262|70<sub>1050</sub>XQN|7×36<sup>1053</sup>+43727|1054|1640|http://factordb.com/index.php?id=1100000003807362444&open=prime|http://factordb.com/index.php?showid=1100000003807362444&base=36|http://factordb.com/index.php?showid=1100000003807362444|http://factordb.com/cert.php?id=1100000003807362444|http://factordb.com/index.php?query=7*36%5E%28n%2B3%29%2B43727&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35263|EB0<sub>1083</sub>UV|515×36<sup>1085</sup>+1111|1087|1692|http://factordb.com/index.php?id=1100000003807362457&open=prime|http://factordb.com/index.php?showid=1100000003807362457&base=36|http://factordb.com/index.php?showid=1100000003807362457|http://factordb.com/cert.php?id=1100000003807362457|http://factordb.com/index.php?query=515*36%5E%28n%2B2%29%2B1111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35264|F02R0<sub>1237</sub>D|699939×36<sup>1238</sup>+13|1242|1933|http://factordb.com/index.php?id=1100000003807362472&open=prime|http://factordb.com/index.php?showid=1100000003807362472&base=36|http://factordb.com/index.php?showid=1100000003807362472|http://factordb.com/cert.php?id=1100000003807362472|http://factordb.com/index.php?query=699939*36%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35265|50<sub>1313</sub>WMN|5×36<sup>1316</sup>+42287|1317|2049|http://factordb.com/index.php?id=1100000003807362473&open=prime|http://factordb.com/index.php?showid=1100000003807362473&base=36|http://factordb.com/index.php?showid=1100000003807362473|http://factordb.com/cert.php?id=1100000003807362473|http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42287&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35266|V0<sub>1328</sub>444B|31×36<sup>1332</sup>+191963|1333|2075|http://factordb.com/index.php?id=1100000003807362474&open=prime|http://factordb.com/index.php?showid=1100000003807362474&base=36|http://factordb.com/index.php?showid=1100000003807362474|http://factordb.com/cert.php?id=1100000003807362474|http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B191963&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35267|SI0<sub>1712</sub>KH|1026×36<sup>1714</sup>+737|1716|2671|http://factordb.com/index.php?id=1100000003807362475&open=prime|http://factordb.com/index.php?showid=1100000003807362475&base=36|http://factordb.com/index.php?showid=1100000003807362475|http://factordb.com/cert.php?id=1100000003807362475|http://factordb.com/index.php?query=1026*36%5E%28n%2B2%29%2B737&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35268|Z0<sub>1714</sub>387|35×36<sup>1717</sup>+4183|1718|2674|http://factordb.com/index.php?id=1100000003807362477&open=prime|http://factordb.com/index.php?showid=1100000003807362477&base=36|http://factordb.com/index.php?showid=1100000003807362477|http://factordb.com/cert.php?id=1100000003807362477|http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B4183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35269|5<sub>1936</sub>3Z|(36<sup>1938</sup>−295)/7|1938|3016|http://factordb.com/index.php?id=1100000003807362478&open=prime|http://factordb.com/index.php?showid=1100000003807362478&base=36|http://factordb.com/index.php?showid=1100000003807362478|http://factordb.com/cert.php?id=1100000003807362478|http://factordb.com/index.php?query=%2836%5E%28n%2B2%29-295%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35270|40<sub>2478</sub>RV|4×36<sup>2480</sup>+1003|2481|3861|http://factordb.com/index.php?id=1100000002634138559&open=prime|http://factordb.com/index.php?showid=1100000002634138559&base=36|http://factordb.com/index.php?showid=1100000002634138559|http://factordb.com/cert.php?id=1100000002634138559|http://factordb.com/index.php?query=4*36%5E%28n%2B2%29%2B1003&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35271|IS0<sub>2684</sub>A0H|676×36<sup>2687</sup>+12977|2689|4185|http://factordb.com/index.php?id=1100000003807362479&open=prime|http://factordb.com/index.php?showid=1100000003807362479&base=36|http://factordb.com/index.php?showid=1100000003807362479|http://factordb.com/cert.php?id=1100000003807362479|http://factordb.com/index.php?query=676*36%5E%28n%2B3%29%2B12977&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35272|5Z<sub>2859</sub>95|6×36<sup>2861</sup>−967|2862|4454|http://factordb.com/index.php?id=1100000003807362480&open=prime|http://factordb.com/index.php?showid=1100000003807362480&base=36|http://factordb.com/index.php?showid=1100000003807362480|http://factordb.com/cert.php?id=1100000003807362480|http://factordb.com/index.php?query=6*36%5E%28n%2B2%29-967&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35273|Q<sub>2942</sub>2D|(26×36<sup>2944</sup>−30721)/35|2944|4582|http://factordb.com/index.php?id=1100000003807362481&open=prime|http://factordb.com/index.php?showid=1100000003807362481&base=36|http://factordb.com/index.php?showid=1100000003807362481|http://factordb.com/cert.php?id=1100000003807362481|http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-30721%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35274|D0<sub>3047</sub>6E01|13×36<sup>3051</sup>+298081|3052|4750|http://factordb.com/index.php?id=1100000003807362482&open=prime|http://factordb.com/index.php?showid=1100000003807362482&base=36|http://factordb.com/index.php?showid=1100000003807362482|http://factordb.com/cert.php?id=1100000003807362482|http://factordb.com/index.php?query=13*36%5E%28n%2B4%29%2B298081&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35275|CNS<sub>3424</sub>J|(2279×36<sup>3425</sup>−49)/5|3427|5333|http://factordb.com/index.php?id=1100000003807362483&open=prime|http://factordb.com/index.php?showid=1100000003807362483&base=36|http://factordb.com/index.php?showid=1100000003807362483|http://factordb.com/cert.php?id=1100000003807362483|http://factordb.com/index.php?query=%282279*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35276|4<sub>3925</sub>V|(4×36<sup>3926</sup>+941)/35|3926|6110|http://factordb.com/index.php?id=1100000002332536659&open=prime|http://factordb.com/index.php?showid=1100000002332536659&base=36|http://factordb.com/index.php?showid=1100000002332536659|http://factordb.com/cert.php?id=1100000002332536659|http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B941%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35277|OZ<sub>3932</sub>AZ|25×36<sup>3934</sup>−901|3935|6124|http://factordb.com/index.php?id=1100000000840634476&open=prime|http://factordb.com/index.php?showid=1100000000840634476&base=36|http://factordb.com/index.php?showid=1100000000840634476|http://factordb.com/cert.php?id=1100000000840634476|http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35278|RY<sub>4562</sub>H|(979×36<sup>4563</sup>−629)/35|4564|7103|http://factordb.com/index.php?id=1100000003807362485&open=prime|http://factordb.com/index.php?showid=1100000003807362485&base=36|http://factordb.com/index.php?showid=1100000003807362485|http://factordb.com/cert.php?id=1100000003807362485|http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35279|T09<sub>4618</sub>1|(36549×36<sup>4619</sup>−289)/35|4621|7192|http://factordb.com/index.php?id=1100000003807362486&open=prime|http://factordb.com/index.php?showid=1100000003807362486&base=36|http://factordb.com/index.php?showid=1100000003807362486|http://factordb.com/cert.php?id=1100000003807362486|http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35280|FZ<sub>5777</sub>3P|16×36<sup>5779</sup>−1163|5780|8996|http://factordb.com/index.php?id=1100000003807362487&open=prime|http://factordb.com/index.php?showid=1100000003807362487&base=36|http://factordb.com/index.php?showid=1100000003807362487|http://factordb.com/cert.php?id=1100000003807362487|http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35281|EO0<sub>6177</sub>V|528×36<sup>6178</sup>+31|6180|9618|http://factordb.com/index.php?id=1100000003807362488&open=prime|http://factordb.com/index.php?showid=1100000003807362488&base=36|http://factordb.com/index.php?showid=1100000003807362488|http://factordb.com/cert.php?id=1100000003807362488|http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35282|VL0<sub>7258</sub>J|1137×36<sup>7259</sup>+19|7261|11301|http://factordb.com/index.php?id=1100000003807362489&open=prime|http://factordb.com/index.php?showid=1100000003807362489&base=36|http://factordb.com/index.php?showid=1100000003807362489|http://factordb.com/cert.php?id=1100000003807362489|http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|
|36|35283|J<sub>10117</sub>LJ|(19×36<sup>10119</sup>+2501)/35|10119|15748|http://factordb.com/index.php?id=1100000003807362491&open=prime|http://factordb.com/index.php?showid=1100000003807362491&base=36|http://factordb.com/index.php?showid=1100000003807362491|http://factordb.com/cert.php?id=1100000003807362491|http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|

Condensed table for bases 2 ≤ *b* ≤ 36: (the bases *b* = 11, 13, 16, 17, 19, 21\~23, 25\~36 data assumes the primality of the probable primes) (This data assumes that a number > 10<sup>25000</sup> which has passed the Miller–Rabin primality tests (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases *p* < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html) and has trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) to 10<sup>16</sup> is in fact prime, since in some cases (e.g. *b* = 11) a candidate for minimal prime base *b* is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 57125 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=134371), see https://t5k.org/top20/page.php?id=27, and 65263 > 57125)

Two coincidences (https://en.wikipedia.org/wiki/Mathematical_coincidence, https://mathworld.wolfram.com/Coincidence.html):

* The length of the largest minimal prime in bases *b* = 18 and *b* = 20 are both 6271 (in theory, *b* = 20 should be about 1.65 times as *b* = 18).
* The length of the 10401st minimal prime in base *b* = 17 is exactly 10401.

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and http://alfredreichlg.de/, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm)

(all small prime factors (< 10<sup>9</sup>, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4\*9^n-1", and it will tell you that this form can be factored to (2×3<sup>*n*</sup>−1) × (2×3<sup>*n*</sup>+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0)) in corresponding families of these (probable) primes were added to *factordb*)

|*b*|number of minimal primes (or probable prime, which is a minimal prime assuming its primality) base *b*|base-*b* form of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base *b* (write "*d*<sub>*n*</sub>" if there are 5 or more (*n*) consecutive same digits *d*)|length of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base *b*|length of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base *b* in decimal|algebraic ((*a*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base *b*|*factordb* entry of this minimal prime (or probable prime, which is a minimal prime assuming its primality)|this minimal prime (or probable prime, which is a minimal prime assuming its primality) written in base *b*|this minimal prime (or probable prime, which is a minimal prime assuming its primality) written in decimal|factorization of the numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b* (if *n* = 0 already makes the number > *b*, then start with *n* = 0))|number of unsolved families in base *b*|searching limit of length for the unsolved families in base *b* (if there are different searching limits for the unsolved families in base *b*, choose the lowest searching limit)|
|---|---|---|---|---|---|---|---|---|---|---|---|
|2|1|11|2|1|3|http://factordb.com/index.php?id=3&open=ecm|http://factordb.com/index.php?showid=3&base=2|http://factordb.com/index.php?showid=3|–|0|–|
|3|3|111<br>21<br>12|3<br>2<br>2|2<br>1<br>1|13<br>7<br>5|http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm|http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3|http://factordb.com/index.php?showid=13<br>http://factordb.com/index.php?showid=7<br>http://factordb.com/index.php?showid=5|–<br>–<br>–|0|–|
|4|5|221<br>31<br>23<br>13<br>11|3<br>2<br>2<br>2<br>2|2<br>2<br>2<br>1<br>1|41<br>13<br>11<br>7<br>5|http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm|http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4|http://factordb.com/index.php?showid=41<br>http://factordb.com/index.php?showid=13<br>http://factordb.com/index.php?showid=11<br>http://factordb.com/index.php?showid=7<br>http://factordb.com/index.php?showid=5|–<br>–<br>–<br>–<br>–|0|–|
|5|22|10<sub>93</sub>13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414|96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3|67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3|5<sup>95</sup>+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109|http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm|http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5|http://factordb.com/index.php?showid=1100000000034686071<br>http://factordb.com/index.php?showid=9391<br>http://factordb.com/index.php?showid=3121<br>http://factordb.com/index.php?showid=2341<br>http://factordb.com/index.php?showid=2251<br>http://factordb.com/index.php?showid=1951<br>http://factordb.com/index.php?showid=1249<br>http://factordb.com/index.php?showid=653<br>http://factordb.com/index.php?showid=401<br>http://factordb.com/index.php?showid=109|http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>–<br>–<br>–<br>–<br>–<br>–<br>–<br>–<br>–|0|–|
|6|11|40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15|5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2|4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2|5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11|http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm|http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6|http://factordb.com/index.php?showid=5209<br>http://factordb.com/index.php?showid=1033<br>http://factordb.com/index.php?showid=1009<br>http://factordb.com/index.php?showid=31<br>http://factordb.com/index.php?showid=29<br>http://factordb.com/index.php?showid=23<br>http://factordb.com/index.php?showid=19<br>http://factordb.com/index.php?showid=17<br>http://factordb.com/index.php?showid=13<br>http://factordb.com/index.php?showid=11|–<br>–<br>–<br>–<br>–<br>–<br>–<br>–<br>–<br>–|0|–|
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|28|25528\~25529|O4O<sub>94535</sub>9<br>5OA<sub>31238</sub>F<br>N6<sub>24051</sub>LR<br>D0<sub>5267</sub>77D<br>QO<sub>4239</sub>69<br>5<sub>3746</sub>8P<br>G0<sub>1899</sub>AN<br>A<sub>1423</sub>6F<br>5I<sub>1370</sub>F<br>5<sub>1332</sub>P8P|94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335|136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932|(6092×28<sup>94536</sup>−143)/9<br>(4438×28<sup>31239</sup>+125)/27<br>(209×28<sup>24053</sup>+3967)/9<br>13×28<sup>5270</sup>+5697<br>(242×28<sup>4241</sup>−4679)/9<br>(5×28<sup>3748</sup>+2803)/27<br>16×28<sup>1901</sup>+303<br>(10×28<sup>1425</sup>−2899)/27<br>(17×28<sup>1371</sup>−11)/3<br>(5×28<sup>1335</sup>+426163)/27|http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime|http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28|http://factordb.com/index.php?showid=1100000000808118231<br>http://factordb.com/index.php?showid=1100000003880455200<br>http://factordb.com/index.php?showid=1100000003879667576<br>http://factordb.com/index.php?showid=1100000003850151420<br>http://factordb.com/index.php?showid=1100000000840839934<br>http://factordb.com/index.php?showid=1100000003850161974<br>http://factordb.com/index.php?showid=1100000003850161973<br>http://factordb.com/index.php?showid=1100000000840839947<br>http://factordb.com/index.php?showid=1100000003850161972<br>http://factordb.com/index.php?showid=1100000003850161965|http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|1|543202|
|29|≥353000|||||||||(still have many non-linear left families)||
|30|2619|OT<sub>34205</sub><br>I0<sub>24608</sub>D<br>5<sub>4882</sub>J<br>C0<sub>1022</sub>1<br>M0<sub>547</sub>SS7<br>M<sub>241</sub>QB<br>AN<sub>206</sub><br>50<sub>164</sub>B<br>J<sub>153</sub>QJ<br>J<sub>94</sub>QQJ|34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97|50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144|25×30<sup>34205</sup>−1<br>18×30<sup>24609</sup>+13<br>(5×30<sup>4883</sup>+401)/29<br>12×30<sup>1023</sup>+1<br>22×30<sup>550</sup>+26047<br>(22×30<sup>243</sup>+3139)/29<br>(313×30<sup>206</sup>−23)/29<br>5×30<sup>165</sup>+11<br>(19×30<sup>155</sup>+6071)/29<br>(19×30<sup>97</sup>+188771)/29|http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm|http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30|http://factordb.com/index.php?showid=1100000000800812865<br>http://factordb.com/index.php?showid=1100000003593967511<br>http://factordb.com/index.php?showid=1100000002327649423<br>http://factordb.com/index.php?showid=1100000000785448736<br>http://factordb.com/index.php?showid=1100000003593407988<br>http://factordb.com/index.php?showid=1100000003593408295<br>http://factordb.com/index.php?showid=1100000002327651073<br>http://factordb.com/index.php?showid=1100000002356282476<br>http://factordb.com/index.php?showid=1100000003593409109<br>http://factordb.com/index.php?showid=1100000003593409165|http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=12*30%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=5*30%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2819*30%5E%28n%2B2%29%2B6071%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2819*30%5E%28n%2B3%29%2B188771%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|0|–|
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|34|184785\~184832|GFGC<sub>99996</sub>5<br>90<sub>97950</sub>FJ<br>NM0<sub>85218</sub>KX<br>F<sub>83189</sub>H2HP<br>P<sub>79441</sub>444P<br>6<sub>77027</sub>8X<br>XQIQ<sub>72241</sub>D<br>T<sub>66530</sub>IF<br>4<sub>66152</sub>B<br>2EEC<sub>66039</sub>7|100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043|153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143|(209246×34<sup>99997</sup>−81)/11<br>9×34<sup>97952</sup>+529<br>804×34<sup>85220</sup>+713<br>(5×34<sup>83193</sup>+700233)/11<br>(25×34<sup>79445</sup>−28062367)/33<br>(2×34<sup>77029</sup>+1043)/11<br>(1288676×34<sup>72242</sup>−455)/33<br>(29×34<sup>66532</sup>−12833)/33<br>(4×34<sup>66153</sup>+227)/33<br>(30826×34<sup>66040</sup>−59)/11|http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime|http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34|http://factordb.com/index.php?showid=1100000004702891268<br>http://factordb.com/index.php?showid=1100000004702894713<br>http://factordb.com/index.php?showid=1100000004702900996<br>http://factordb.com/index.php?showid=1100000004702910376<br>http://factordb.com/index.php?showid=1100000004702913746<br>http://factordb.com/index.php?showid=1100000004702918600<br>http://factordb.com/index.php?showid=1100000004399656529<br>http://factordb.com/index.php?showid=1100000004399657696<br>http://factordb.com/index.php?showid=1100000004399658651<br>http://factordb.com/index.php?showid=1100000004399659716|http://factordb.com/index.php?query=%28209246*34%5E%28n%2B1%29-81%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=9*34%5E%28n%2B2%29%2B529&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=804*34%5E%28n%2B2%29%2B713&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%285*34%5E%28n%2B4%29%2B700233%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2825*34%5E%28n%2B4%29-28062367%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%282*34%5E%28n%2B2%29%2B1043%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%281288676*34%5E%28n%2B1%29-455%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2829*34%5E%28n%2B2%29-12833%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%284*34%5E%28n%2B1%29%2B227%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2830826*34%5E%28n%2B1%29-59%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|47|100000|
|35|≥633000|||||||||(still have many non-linear left families)||
|36|35286\~35290|P<sub>81993</sub>SZ<br>S0<sub>75007</sub>8H<br>7K<sub>26567</sub>Z<br>J<sub>10117</sub>LJ<br>VL0<sub>7258</sub>J<br>EO0<sub>6177</sub>V<br>FZ<sub>5777</sub>3P<br>T09<sub>4618</sub>1<br>RY<sub>4562</sub>H<br>OZ<sub>3932</sub>AZ|81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935|127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124|(5×36<sup>81995</sup>+821)/7<br>28×36<sup>75009</sup>+305<br>(53×36<sup>26568</sup>+101)/7<br>(19×36<sup>10119</sup>+2501)/35<br>1137×36<sup>7259</sup>+19<br>528×36<sup>6178</sup>+31<br>16×36<sup>5779</sup>−1163<br>(36549×36<sup>4619</sup>−289)/35<br>(979×36<sup>4563</sup>−629)/35<br>25×36<sup>3934</sup>−901|http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime|http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36|http://factordb.com/index.php?showid=1100000002394962083<br>http://factordb.com/index.php?showid=1100000004020085177<br>http://factordb.com/index.php?showid=1100000003896952461<br>http://factordb.com/index.php?showid=1100000003807362491<br>http://factordb.com/index.php?showid=1100000003807362489<br>http://factordb.com/index.php?showid=1100000003807362488<br>http://factordb.com/index.php?showid=1100000003807362487<br>http://factordb.com/index.php?showid=1100000003807362486<br>http://factordb.com/index.php?showid=1100000003807362485<br>http://factordb.com/index.php?showid=1100000000840634476|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show<br>http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show|4|100000|

These "unsolved" families in fact have larger (probable) primes, but since the length of these (probable) primes are larger than the search limit of the corresponding bases *b*, and they are not in the kernel files (nor in the condensed table) since they *may* not be the next minimal primes in base *b*, and the indices of these minimal primes in base *b* are unknown:

* Family 4{9} in base *b* = 17, its algebraic form is (73×17<sup>*n*</sup>−9)/16 with *n* ≥ 1, which has a probable prime at length 111334 (this probable prime is 49<sub>111333</sub>), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 17, the formula of this probable prime is (73×17<sup>111333</sup>−9)/16, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%2873*17%5E111333-9%29%2F16&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118219&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.17.txt
* Family 97{0}1 in base *b* = 17, its algebraic form is 160×17<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 166050 (this prime is 970<sub>166047</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 17, the formula of this prime is 160×17<sup>166048</sup>+1, the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000000765961411&open=prime, see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17
* Family F7{0}1 in base *b* = 17, its algebraic form is 262×17<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 186770 (this prime is F70<sub>186767</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 17, the formula of this prime is 262×17<sup>186768</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=85256, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000000765961429&open=prime, see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17
* Family 1E7{0}1 in base *b* = 19, its algebraic form is 634×19<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 122899 (this prime is 1E70<sub>122896</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 19, the formula of this prime is 634×19<sup>122897</sup>+1, the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000001582289581&open=prime, see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19
* Family FG{6} in base *b* = 19, its algebraic form is (904×19<sup>*n*</sup>−1)/3 with *n* ≥ 0, which has a probable prime at length 110986 (this probable prime is FG6<sub>110984</sub>), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 19, the formula of this probable prime is (904×19<sup>110984</sup>−1)/3, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%28904*19%5E110984-1%29%2F3&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118212&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.19.txt
* Family C{F}0K in base *b* = 21, its algebraic form is (51×21<sup>*n*</sup>−1243)/4 with *n* ≥ 0, which has a probable prime at length 479150 (this probable prime is CF<sub>479147</sub>0K), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 21, the formula of this probable prime is (51×21<sup>479149</sup>−1243)/4, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%2851*21%5E479149-1243%29%2F4&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000805209046&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.21.txt
* Family 8{0}1 in base *b* = 23, its algebraic form is 8×23<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 119216 (this prime is 80<sub>119214</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 23, the formula of this prime is 8×23<sup>119215</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=85951, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000000634720609&open=prime, see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S23 (since this family covers another unsolved family 8{0}81, thus the family 8{0}81 only needs to search to length 119216)
* Family 9{E} in base *b* = 23, its algebraic form is (106×23<sup>*n*</sup>−7)/11 with *n* ≥ 1, which has a probable prime at length 800874 (this probable prime is 9E<sub>800873</sub>), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 23, the formula of this probable prime is (106×23<sup>800873</sup>−7)/11, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000782858648&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.23.txt
* Family 1J71{0}1 in base *b* = 25, its algebraic form is 27676×25<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 96277 (this prime is 1J710<sub>96272</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 5 (27676×5<sup>*r*</sup>+1 can be prime only if *r* is even, thus can be converted to 27676×25<sup>*r*/2</sup>+1), the formula of this prime is 27676×25<sup>96273</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=79013, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000003983674902&open=prime, see https://www.mersenneforum.org/showpost.php?p=94583&postcount=18
* Family 9{6}M in base *b* = 25, its algebraic form is (37×25<sup>*n*+1</sup>+63)/4 with *n* ≥ 0, which has a probable prime at length 136967 (this probable prime is 96<sub>136965</sub>M), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 25, the formula of this probable prime is (37×25<sup>136966</sup>+63)/4, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%2837*25%5E136966%2B63%29%2F4&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118185&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt
* Family 71JD{0}1 in base *b* = 25, its algebraic form is 110488×25<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 458554 (this prime is 71JD0<sub>458549</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 5 (110488×5<sup>*r*</sup>+1 can be prime only if *r* is even, thus can be converted to 110488×25<sup>*r*/2</sup>+1), the formula of this prime is 110488×25<sup>458550</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=111834, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000002341496334&open=prime, see http://www.primegrid.com/forum_thread.php?id=5087
* Family DKJ{0}1 in base *b* = 25, its algebraic form is 8644×25<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 246812 (this prime is DKJ0<sub>246808</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 5 (8644×5<sup>*r*</sup>+1 can be prime only if *r* is even, thus can be converted to 8644×25<sup>*r*/2</sup>+1), the formula of this prime is 8644×25<sup>246809</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=94113, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000003983678207&open=prime, see https://www.mersenneforum.org/showpost.php?p=224809&postcount=24
* Family {E}FOO in base *b* = 25, its algebraic form is (7×25<sup>*n*+3</sup>+10613)/12 with *n* ≥ 0, which has a probable prime at length 98399 (this probable prime is E<sub>98396</sub>FOO), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 25, the formula of this probable prime is (7×25<sup>98399</sup>+10613)/12, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%287*25%5E98399%2B10613%29%2F12&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118215&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt
* Family M{F}0F6 in base *b* = 25, its algebraic form is (181×25<sup>*n*+3</sup>−75077)/8 with *n* ≥ 0, which has a probable prime at length 109992 (this probable prime is M<sub>109988</sub>0F6), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 25, the formula of this probable prime is (181×25<sup>109991</sup>−75077)/8, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%28181*25%5E109991-75077%29%2F8&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118206&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt
* Family A{0}PM in base *b* = 27, its algebraic form is 10×27<sup>*n*+2</sup>+697 with *n* ≥ 0, which has a probable prime at length 109006 (this probable prime is A0<sub>109003</sub>PM), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 27, the formula of this probable prime is 10×27<sup>109005</sup>+697, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=10*27%5E109005%2B697&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118203&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt
* Family CA0{F}A in base *b* = 27, its algebraic form is (234483×27<sup>*n*+1</sup>−145)/26 with *n* ≥ 0, which has a probable prime at length 88887 (this probable prime is CA0F<sub>88883</sub>A), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 27, the formula of this probable prime is (234483×27<sup>88884</sup>−145)/26, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%28234483*27%5E88884-145%29%2F26&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118233&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt
* Family {L}G in base *b* = 27, its algebraic form is (21×27<sup>*n*+1</sup>−151)/26 with *n* ≥ 1, which has a probable prime at length 101106 (this probable prime is L<sub>101105</sub>G), this probable prime is found in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 27, the formula of this probable prime is (21×27<sup>101106</sup>−151)/26, the entry of this probable prime in top probable primes is http://www.primenumbers.net/prptop/searchform.php?form=%2821*27%5E101106-151%29%2F26&action=Search, and the entry of this probable prime in *factordb* is http://factordb.com/index.php?id=1100000000808118209&open=prime, see https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt (since this family covers two other unsolved families {L}0G and N9{L}G, thus the family {L}0G only needs to search to length 101106 and the family N9{L}G only needs to search to length 101107)
* Family NU{0}1 in base *b* = 32, its algebraic form is 766×32<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 661866 (this prime is NU0<sub>661863</sub>1), this prime is found in the process of finding the Proth primes *k*×2<sup>*r*</sup>+1 for *k* = 383, the formula of this prime is 766×32<sup>661864</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=134216, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000003813355148&open=prime, see http://www.prothsearch.com/riesel1a.html, 3309321 is the smallest exponent *r* == 1 mod 5 (≥ 6) for *k* = 383 (since 766×32<sup>*n*+1</sup>+1 = 383×2<sup>5×(*n*+1)+1</sup>+1, thus we need an exponent *r* == 1 mod 5 for 383×2<sup>*r*</sup>+1, i.e. the Proth number for *k* = 383, and since *n* ≥ 0, 5×(*n*+1)+1 must be ≥ 5×1+1 = 6)
* Family N7{0}1 in base *b* = 33, its algebraic form is 766×33<sup>*n*+1</sup>+1 with *n* ≥ 0, which has a prime at length 610414 (this prime is N70<sub>610411</sub>1), this prime is found in the process of solving the Sierpinski conjecture in base *b* = 33, the formula of this prime is 766×33<sup>610412</sup>+1, the entry of this prime in top primes is https://t5k.org/primes/page.php?id=121575, and the entry of this prime in *factordb* is http://factordb.com/index.php?id=1100000000838755581&open=prime, see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33

These unsolved families in fact have larger search limit of lengths than the search limit of the corresponding bases *b*:

* Family F1{9} in base *b* = 17, its algebraic form is (4105×17<sup>*n*</sup>−9)/16 with *n* ≥ 0, its search limit is length 1000000, this family is searched in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 17, see https://github.com/curtisbright/mepn-data/blob/master/data/sieve.17.txt
* Family EE1{6} in base *b* = 19, its algebraic form is (15964×19<sup>*n*</sup>−1)/3 with *n* ≥ 0, its search limit is length 707350, this family is searched in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 19, see https://github.com/curtisbright/mepn-data/blob/master/data/sieve.19.txt
* Family G{0}FK in base *b* = 21, its algebraic form is 16×21<sup>*n*+2</sup>+335 with *n* ≥ 0, its search limit is length 506722, this family is searched in the process of solving the original minimal prime problem (i.e. prime > *b* is not required) in base *b* = 21, see https://github.com/curtisbright/mepn-data/blob/master/data/sieve.21.txt
* Family H3{0}1 in base *b* = 23, its algebraic form is 394×23<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 700000, this family is searched in the process of solving the Sierpinski conjecture in base *b* = 529 (394×23<sup>*r*</sup>+1 can be prime only if *r* is even, thus can be converted to 394×529<sup>*r*/2</sup>+1), see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529
* Family JH{0}1 in base *b* = 23, its algebraic form is 454×23<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 700000, this family is searched in the process of solving the Sierpinski conjecture in base *b* = 529 (454×23<sup>*r*</sup>+1 can be prime only if *r* is even, thus can be converted to 454×529<sup>*r*/2</sup>+1), see http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529
* Family D71J{0}1 in base *b* = 25, its algebraic form is 207544×25<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 350000, this family is searched in the process of solving the Sierpinski conjecture in base *b* = 25, see http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm
* Family EF{O} in base *b* = 25, its algebraic form is 366×25<sup>*n*</sup>−1 with *n* ≥ 0, its search limit is length 300000, this family is searched in the process of solving the Riesel conjecture in base *b* = 25, see http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm
* Family {F}G in base *b* = 31, its algebraic form is (31<sup>*n*+1</sup>+1)/2 with *n* ≥ 1, its search limit is length 16777215, this family is searched in the process of finding the generalized half Fermat primes (*b*<sup>2<sup>*r*</sup></sup>+1)/2 in base *b* = 31 ((*b*<sup>*n*+1</sup>+1)/2 can be prime only if *n*+1 is power of 2, thus can be converted to (*b*<sup>2<sup>*r*</sup></sup>+1)/2), for *n*+1 ≤ 524288 see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, and the numbers with *n*+1 = 1048576, 2097152, 4194304, 8388608 are divisible by 37339791361, 5138022401, 998244353, 2121143418881, respectively (these four prime factors are found by trial division), and thus the smallest possible prime is *n*+1 = 2<sup>24</sup> = 16777216, and its length is 16777216
* Family 4{0}1 in base *b* = 32, its algebraic form is 4×32<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 1717986918, this family is searched in the process of finding the Fermat primes 2<sup>2<sup>*s*</sup></sup>+1 (4×32<sup>*n*+1</sup>+1 = 2<sup>5×*n*+7</sup>+1, and 2<sup>5×*n*+7</sup>+1 can be prime only if 5×*n*+7 is a power of 2, thus can be converted to 2<sup>2<sup>*s*</sup></sup>+1), see http://www.prothsearch.com/fermat.html, 2<sup>*s*</sup> == 2 mod 5 if and only if *s* == 1 mod 4, and the smallest *s* == 1 mod 4 (and *s* > 2) such that 2<sup>2<sup>*s*</sup></sup>+1 *may* be prime is *s* = 33, and thus the smallest possible prime is *n*+1 = (2<sup>33</sup>−2)/5 = 1717986918, and its length is 1717986919 (since 4×32<sup>*n*+1</sup>+1 = 2<sup>5×(*n*+1)+2</sup>+1, thus we need an exponent *r* == 2 mod 5 for 2<sup>*r*</sup>+1 = 2<sup>2<sup>*s*</sup></sup>+1 (if 2<sup>*r*</sup>+1 is prime, then *r* is a power of 2, thus we can let *r* = 2<sup>*s*</sup>), and 2<sup>*s*</sup> == 2 mod 5 if and only if *s* == 1 mod 4, and since *n* ≥ 0, 5×(*n*+1)+2 must be ≥ 5×1+2 = 7, thus *s* = *log*<sub>2</sub>(5×(*n*+1)+2) must be > 2)
* Family G{0}1 in base *b* = 32, its algebraic form is 16×32<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 3435973836, this family is searched in the process of finding the Fermat primes 2<sup>2<sup>*s*</sup></sup>+1 (16×32<sup>*n*+1</sup>+1 = 2<sup>5×*n*+9</sup>+1, and 2<sup>5×*n*+9</sup>+1 can be prime only if 5×*n*+9 is a power of 2, thus can be converted to 2<sup>2<sup>*s*</sup></sup>+1), see http://www.prothsearch.com/fermat.html, 2<sup>*s*</sup> == 4 mod 5 if and only if *s* == 2 mod 4, and the smallest *s* == 2 mod 4 (and *s* > 3) such that 2<sup>2<sup>*s*</sup></sup>+1 *may* be prime is *s* = 34, and thus the smallest possible prime is *n*+1 = (2<sup>34</sup>−4)/5 = 3435973836, and its length is 3435973837 (since 16×32<sup>*n*+1</sup>+1 = 2<sup>5×(*n*+1)+4</sup>+1, thus we need an exponent *r* == 4 mod 5 for 2<sup>*r*</sup>+1 = 2<sup>2<sup>*s*</sup></sup>+1 (if 2<sup>*r*</sup>+1 is prime, then *r* is a power of 2, thus we can let *r* = 2<sup>*s*</sup>), and 2<sup>*s*</sup> == 4 mod 5 if and only if *s* == 2 mod 4, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9, thus *s* = *log*<sub>2</sub>(5×(*n*+1)+4) must be > 3)
* Family NG{0}1 in base *b* = 32, its algebraic form is 752×32<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 1800000, this family is searched in the process of finding the Proth primes *k*×2<sup>*r*</sup>+1 for *k* = 47, see http://www.prothsearch.com/riesel1.html, *k* = 47 is searched to exponent 9000000 with no exponent == 4 mod 5 (≥ 9) has been found (since 752×32<sup>*n*+1</sup>+1 = 47×2<sup>5×(*n*+1)+4</sup>+1, thus we need an exponent *r* == 4 mod 5 for 47×2<sup>*r*</sup>+1, i.e. the Proth number for *k* = 47, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9)
* Family UG{0}1 in base *b* = 32, its algebraic form is 976×32<sup>*n*+1</sup>+1 with *n* ≥ 0, its search limit is length 800000, this family is searched in the process of finding the Proth primes *k*×2<sup>*r*</sup>+1 for *k* = 61, see http://www.prothsearch.com/riesel1.html, *k* = 61 is searched to exponent 4000000 with no exponent == 4 mod 5 (≥ 9) has been found (since 976×32<sup>*n*+1</sup>+1 = 61×2<sup>5×(*n*+1)+4</sup>+1, thus we need an exponent *r* == 4 mod 5 for 61×2<sup>*r*</sup>+1, i.e. the Proth number for *k* = 61, and since *n* ≥ 0, 5×(*n*+1)+4 must be ≥ 5×1+4 = 9)
* Family S{V} in base *b* = 32, its algebraic form is 29×32<sup>*n*</sup>−1 with *n* ≥ 1, its search limit is length 2000000, this family is searched in the process of solving the Riesel conjecture in base *b* = 1024 (29×32<sup>*r*</sup>−1 can be prime only if *r* is even, thus can be converted to 29×1024<sup>*r*/2</sup>−1), see http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024
* Family {V}3 in base *b* = 32, its algebraic form is 32<sup>*n*+1</sup>−29 with *n* ≥ 1, its search limit is length 68000, this family is searched in the process of finding the dual Riesel primes 2<sup>*r*</sup>−*k* for *k* = 29, see https://oeis.org/A356826, *k* = 29 is searched to exponent 340000 with no exponent == 0 mod 5 (≥ 10) has been found (since 32<sup>*n*+1</sup>−29 = 2<sup>5×(*n*+1)</sup>−29, thus we need an exponent *r* == 0 mod 5 for 2<sup>*r*</sup>−29, i.e. the dual Riesel number for *k* = 29, and since *n* ≥ 1, 5×(*n*+1) must be ≥ 5×2 = 10)

Links for top (probable) primes: (also pages for the largest known prime: https://en.wikipedia.org/wiki/Largest_known_prime_number, https://en.wikipedia.org/wiki/List_of_largest_known_primes_and_probable_primes, http://www.numericana.com/answer/primes.htm#history, and related pages: https://en.wikipedia.org/wiki/Megaprime, https://t5k.org/glossary/xpage/TitanicPrime.html, https://t5k.org/glossary/xpage/GiganticPrime.html, https://t5k.org/glossary/xpage/Megaprime.html, https://www.rieselprime.de/ziki/Titanic_prime, https://www.rieselprime.de/ziki/Gigantic_prime, https://www.rieselprime.de/ziki/Megaprime, https://www.rieselprime.de/ziki/Gigaprime, https://mathworld.wolfram.com/TitanicPrime.html, https://mathworld.wolfram.com/GiganticPrime.html)

* https://t5k.org/primes/lists/all.txt (top definitely primes)
* https://t5k.org/primes/lists/all.zip (top definitely primes, zipped file)
* https://t5k.org/primes/lists/short.txt (definitely primes with ≥ 800000 decimal digits)
* https://t5k.org/primes/search.php?OnList=all&Number=1000000&Style=HTML (all numbers in the list of top definitely primes, html version)
* https://t5k.org/primes/search.php?OnList=all&Number=1000000 (all numbers in the list of top definitely primes, text version)
* https://t5k.org/primes/ (index page of top definitely primes)
* https://t5k.org/primes/download.php (download page of top definitely primes)
* https://t5k.org/primes/status.php (recently 3 days found top definitely primes)
* https://t5k.org/primes/status.php?hours=1000 (recently found top definitely primes)
* https://t5k.org/primes/status.php?hours=0 (top definitely primes which are in process or awaiting verification)
* https://t5k.org/largest.html (the information page of top definitely primes)
* https://t5k.org/notes/by_year.html (the information page of the largest known prime by year)
* https://t5k.org/notes/faq/why.html (the information page of why do people find these large primes)
* https://t5k.org/primes/search.php (search page of top definitely primes)
* https://t5k.org/primes/search.php?Advanced=1 (advanced search page of top definitely primes)
* https://t5k.org/primes/search_proth.php (search page of top definitely primes of the form *k*×*b*<sup>*n*</sup>±1)
* https://t5k.org/primes/status.php (verification status page of top definitely primes)
* https://t5k.org/top20/index.php (the top 20 definitely primes of certain selected forms)
* https://t5k.org/bios/submission.php (submit page of top definitely primes)
* https://t5k.org/bios/newprover.php (submit page of top definitely primes, create a new prover account)
* https://t5k.org/bios/newcode.php (submit page of top definitely primes, create a new prover code)
* https://t5k.org/bios/index.php (index of the provers and programs and projects of top definitely primes)
* http://www.primenumbers.net/prptop/prptop.php (top probable primes)
* http://www.primenumbers.net/prptop/latest.php (recently found top probable primes)
* http://www.primenumbers.net/prptop/searchform.php (search page of top probable primes)
* http://www.primenumbers.net/prptop/searchform.php?form=%3F&action=Search (all numbers in the list of top probable primes)
* http://www.primenumbers.net/prptop/submit.php (submit page of top probable primes)
* http://www.primenumbers.net/prptop/topdisc.php (index of the provers of top probable primes)

and the search result page for the (probable) primes of special forms: (note: a large prime of the form (*a*×*b*<sup>*n*</sup>+*c*)/*d* with small *a*, *b*, *c*, *d* and large *n* can be easily proven prime if and only if *c* = ±1 and *d* = 1)

Definitely primes (i.e. *c* = ±1 and *d* = 1):

* *b*<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=%5E[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *b*<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=%5E[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×*b*<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×*b*<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×2<sup>*n*</sup>+1 (which includes *a*×4<sup>*n*</sup>+1, *a*×8<sup>*n*</sup>+1, *a*×16<sup>*n*</sup>+1, *a*×32<sup>*n*</sup>+1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×2<sup>*n*</sup>−1 (which includes *a*×4<sup>*n*</sup>−1, *a*×8<sup>*n*</sup>−1, *a*×16<sup>*n*</sup>−1, *a*×32<sup>*n*</sup>−1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*2%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×3<sup>*n*</sup>+1 (which includes *a*×9<sup>*n*</sup>+1, *a*×27<sup>*n*</sup>+1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×3<sup>*n*</sup>−1 (which includes *a*×9<sup>*n*</sup>−1, *a*×27<sup>*n*</sup>−1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*3%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×5<sup>*n*</sup>+1 (which includes *a*×25<sup>*n*</sup>+1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×5<sup>*n*</sup>−1 (which includes *a*×25<sup>*n*</sup>−1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*5%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×6<sup>*n*</sup>+1 (which includes *a*×36<sup>*n*</sup>+1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×6<sup>*n*</sup>−1 (which includes *a*×36<sup>*n*</sup>−1): https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*6%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×7<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×7<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*7%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×10<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×10<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*10%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×11<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×11<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*11%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×12<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×12<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*12%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×13<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×13<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*13%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×14<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×14<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*14%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×15<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×15<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*15%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×17<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×17<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*17%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×18<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×18<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*18%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×19<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×19<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*19%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×20<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×20<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*20%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×21<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×21<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*21%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×22<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×22<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*22%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×23<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×23<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*23%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×24<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×24<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*24%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×26<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×26<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*26%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×28<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×28<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*28%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×29<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×29<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*29%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×30<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×30<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*30%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×31<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×31<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*31%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×33<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×33<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*33%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×34<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×34<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*34%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×35<sup>*n*</sup>+1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML
* *a*×35<sup>*n*</sup>−1: https://t5k.org/primes/search.php?Description=[[:digit:]]%7B1,%7D*35%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML
* *a*×2<sup>*n*</sup>±1 (which includes *a*×4<sup>*n*</sup>±1, *a*×8<sup>*n*</sup>±1, *a*×16<sup>*n*</sup>±1, *a*×32<sup>*n*</sup>±1): https://t5k.org/primes/search_proth.php?base=2&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×3<sup>*n*</sup>±1 (which includes *a*×9<sup>*n*</sup>±1, *a*×27<sup>*n*</sup>±1): https://t5k.org/primes/search_proth.php?base=3&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×5<sup>*n*</sup>±1 (which includes *a*×25<sup>*n*</sup>±1): https://t5k.org/primes/search_proth.php?base=5&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×6<sup>*n*</sup>±1 (which includes *a*×36<sup>*n*</sup>±1): https://t5k.org/primes/search_proth.php?base=6&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×7<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=7&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×10<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=10&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×11<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=11&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×12<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=12&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×13<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=13&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×14<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=14&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×15<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=15&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×17<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=17&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×18<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=18&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×19<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=19&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×20<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=20&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×21<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=21&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×22<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=22&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×23<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=23&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×24<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=24&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×26<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=26&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×28<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=28&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×29<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=29&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×30<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=30&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×31<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=31&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×33<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=33&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×34<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=34&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search
* *a*×35<sup>*n*</sup>±1: https://t5k.org/primes/search_proth.php?base=35&min_k=&max_k=&min_n=&max_n=&plus=on&minus=on&number=1000000&search=Start+Search

Probable primes (i.e. *c* ≠ ±1 or/and *d* ≠ 1):

* *b*<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2Bc&action=Search
* *b*<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=b%5En-c&action=Search
* *a*×*b*<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En%2Bc&action=Search
* *a*×*b*<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En-c&action=Search
* (*b*<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En%2Bc%29%2Fd&action=Search
* (*b*<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-c%29%2Fd&action=Search
* (*a*×*b*<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En%2Bc%29%2Fd&action=Search
* (*a*×*b*<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En-c%29%2Fd&action=Search

* 2<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=2%5En%2Bc&action=Search
* 2<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=2%5En-c&action=Search
* *a*×2<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*2%5En%2Bc&action=Search
* *a*×2<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*2%5En-c&action=Search
* (2<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%282%5En%2Bc%29%2Fd&action=Search
* (2<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%282%5En-c%29%2Fd&action=Search
* (*a*×2<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*2%5En%2Bc%29%2Fd&action=Search
* (*a*×2<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*2%5En-c%29%2Fd&action=Search
* 3<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=3%5En%2Bc&action=Search
* 3<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=3%5En-c&action=Search
* *a*×3<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*3%5En%2Bc&action=Search
* *a*×3<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*3%5En-c&action=Search
* (3<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%283%5En%2Bc%29%2Fd&action=Search
* (3<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%283%5En-c%29%2Fd&action=Search
* (*a*×3<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*3%5En%2Bc%29%2Fd&action=Search
* (*a*×3<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*3%5En-c%29%2Fd&action=Search
* 10<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=10%5En%2Bc&action=Search
* 10<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=10%5En-c&action=Search
* *a*×10<sup>*n*</sup>+*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*10%5En%2Bc&action=Search
* *a*×10<sup>*n*</sup>−*c*: http://www.primenumbers.net/prptop/searchform.php?form=a*10%5En-c&action=Search
* (10<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%2810%5En%2Bc%29%2Fd&action=Search
* (10<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%2810%5En-c%29%2Fd&action=Search
* (*a*×10<sup>*n*</sup>+*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*10%5En%2Bc%29%2Fd&action=Search
* (*a*×10<sup>*n*</sup>−*c*)/*d*: http://www.primenumbers.net/prptop/searchform.php?form=%28a*10%5En-c%29%2Fd&action=Search

Home page of Proth Primes Search (search of primes of the form *k*×2<sup>*n*</sup>+1 with odd *k*): http://www.prothsearch.com/

Home page of Riesel Prime Search (search of primes of the form *k*×2<sup>*n*</sup>−1 with odd *k*): https://web.archive.org/web/20210817181915/http://www.15k.org/

References of minimal primes (https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics), https://t5k.org/glossary/xpage/MinimalPrime.html) (the original definition, i.e. prime > base (*b*) is not required):

1. http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) (base 10)
2. https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) (bases 2 to 30)
3. https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) (bases 2 to 30)
4. https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf) (bases 2 to 30)
5. https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf) (bases 2 to 30)
6. https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) (bases 2 to 10) **(warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of *S*<sub>***m***</sub> for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively)**
7. https://github.com/curtisbright/mepn-data (bases 2 to 30)
8. https://github.com/curtisbright/mepn (bases 2 to 30)
9. https://github.com/RaymondDevillers/primes (bases 28 to 50)
10. http://recursed.blogspot.com/2006/12/prime-game.html (base 10)
11. https://inzitan.blogspot.com/2007/07/prime-game.html (in Spain) (base 10)
12. http://www.pourlascience.fr/ewb_pages/a/article-nombres-premiers-inevitables-et-pyramidaux-24978.php (in French) (base 10)
13. http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremInev.htm (base 10)
14. https://schoolbag.info/mathematics/numbers/66.html (base 10)
15. https://www.microsiervos.com/archivo/ciencia/2-3-5-7-11.html (in Spain) (base 10)
16. https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it (base 10)
17. https://www.metafilter.com/62794/3-is-an-odd-prime-5-is-an-odd-prime-7-is-an-odd-prime-9-is-a-very-odd-prime (base 10)
18. https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1165031124 (base 10)
19. https://www.cristal.univ-lille.fr/profil/jdelahay/pls:2002:094.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_100.pdf) (bases 2 to 10) **(warning: the data for base 8 has error, the data misses the primes 444641 and 444444441)**
20. https://logs.esolangs.org/freenode-esoteric/2011-02-04.html (bases 2 to 10) **(warning: the data for base 8 has error, the data misses the prime 111 and instead wrongly includes the primes 1101, 101111, 600111, 1000011, 1000111, 4411111, 64111111, 601111111, 41111111111111111, and possibly 6111111111111111111111 if the author of this article continues to search)**
21. http://www.bitman.name/math/article/730 (in Italian) (bases 2 to 20)
22. http://www.bitman.name/math/table/497 (in Italian) (bases 2 to 16)
23. http://www.bitman.name/math/table/498 (in Italian) (base 17)
24. http://www.bitman.name/math/table/499 (in Italian) (base 18)
25. http://www.bitman.name/math/table/500 (in Italian) (base 19)
26. http://www.bitman.name/math/table/501 (in Italian) (base 20)
27. https://www.primepuzzles.net/puzzles/puzz_178.htm (base 10)
28. https://oeis.org/A071062 (base 10)

Other researches for the digits of the primes:

Left-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty suffix is prime:

1. http://primerecords.dk/left-truncatable.txt (base 10)
2. http://chesswanks.com/num/LTPs/ (bases 3 to 120)
3. https://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base (bases 3 to 17)
4. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 to 11)
5. http://www.primerecords.dk/left-truncatable.htm (base 10)
6. http://rosettacode.org/wiki/Truncatable_primes (base 10)
7. https://www.primepuzzles.net/puzzles/puzz_002.htm (base 10)
8. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base 10)
9. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 to 20)
10. http://www.bitman.name/math/table/524 (in Italian) (bases 2 to 20)
11. https://oeis.org/A103443 (largest left-truncatable prime in base *b*)
12. https://oeis.org/A103463 (length of the largest left-truncatable prime in base *b*)
13. https://oeis.org/A076623 (number of left-truncatable primes in base *b*)

Right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty prefix is prime:

1. http://primerecords.dk/right-truncatable.txt (base 10)
2. http://fatphil.org/maths/rtp/rtp.html (bases 3 to 90)
3. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 to 15)
4. http://rosettacode.org/wiki/Truncatable_primes (base 10)
5. https://www.primepuzzles.net/puzzles/puzz_002.htm (base 10)
6. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base 10)
7. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 to 20)
8. http://www.bitman.name/math/table/525 (in Italian) (bases 2 to 20)
9. https://oeis.org/A023107 (largest right-truncatable prime in base *b*)
10. https://oeis.org/A103483 (length of the largest right-truncatable prime in base *b*)
11. https://oeis.org/A076586 (number of right-truncatable primes in base *b*)

Other researches for the minimal elements of other subsets of positive integers written in the positional numeral system with radix *b*, as digit strings with subsequence ordering:

Primes == 1 mod 4:

1. https://www.primepuzzles.net/puzzles/puzz_178.htm
2. https://github.com/curtisbright/mepn-data/blob/master/data/primes1mod4/minimal.10.txt
3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) **(warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4*k*+1 prime" in the list: 9<sub>**630**</sub>493, this prime is not a minimal 4*k*+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9<sub>**630**</sub>493, there are 146 (instead of 173) minimal 4*k*+1 primes and 113 (instead of 138) minimal 4*k*−1 primes, and the largest minimal 4*k*+1 prime is 8<sub>**77**</sub>33 = (8*10<sup>**79**</sup>−503)/9 instead of 9<sub>**630**</sub>493 = 10<sup>**633**</sup>−507)**
4. https://oeis.org/A111055

Primes == 3 mod 4:

1. https://www.primepuzzles.net/puzzles/puzz_178.htm
2. https://github.com/curtisbright/mepn-data/blob/master/data/primes3mod4/minimal.10.txt
3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) **(warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4*k*+1 prime" in the list: 9<sub>**630**</sub>493, this prime is not a minimal 4*k*+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9<sub>**630**</sub>493, there are 146 (instead of 173) minimal 4*k*+1 primes and 113 (instead of 138) minimal 4*k*−1 primes, and the largest minimal 4*k*+1 prime is 8<sub>**77**</sub>33 = (8*10<sup>**79**</sup>−503)/9 instead of 9<sub>**630**</sub>493 = 10<sup>**633**</sup>−507)**
4. https://oeis.org/A111056 **(warning: the b-file does not include the prime 2<sub>**19151**</sub>99)**

Palindromic primes:

1. https://www.primepuzzles.net/puzzles/puzz_178.htm
2. https://oeis.org/A114835 **(warning: the b-file does not include the probable prime 994<sub>**34019**</sub>99)**

Composites:

1. https://github.com/curtisbright/mepn-data/tree/master/data/composites
2. http://www.bitman.name/math/table/504
3. https://oeis.org/A071070

Squares:

1. http://recursed.blogspot.com/2006/12/prime-game.html
2. https://oeis.org/A130448

Powers of 2:

1. https://oeis.org/A071071/a071071.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_13.pdf)
2. https://oeis.org/A071071

Multiples of 3:

1. https://oeis.org/A071073

Multiples of 4:

1. https://oeis.org/A071072

Other sets:

1. https://arxiv.org/pdf/1607.01548.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_14.pdf) (sums of three squares, quadratic residues mod 6, quadratic residues mod 7, range of Euler's totient function, range of "Euler's totient function + 3", range of Dedekind psi function, perfect numbers)
2. https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_15.pdf) (range of "Euler's totient function + *n*", for 0 ≤ *n* ≤ 5)

|set (base *b* = 10)|the set of the minimal elements under the subsequence ordering|number of such elements|length of the longest such element|
|---|---|---|---|
|primes == 1 mod 4|5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, 11969, 19121, 20021, 20201, 21121, 23021, 23201, 43669, 44777, 47777, 60493, 60649, 66749, 80833, 90121, 91121, 91921, 91969, 94693, 111121, 112121, 119921, 199921, 220301, 466369, 470077, 666493, 666649, 772721, 777221, 777781, 779981, 799921, 800333, 803333, 806033, 833033, 833633, 860333, 863633, 901169, 946369, 946669, 999169, 1111169, 1999969, 4007077, 4044077, 4400477, 4666693, 8000033, 8000633, 8006633, 8600633, 8660033, 8830033, 8863333, 8866633, 22000001, 40400077, 44040077, 60000049, 66000049, 66600049, 79999981, 80666633, 83333333, 86606633, 86666633, 88600033, 88883033, 88886033, 400000477, 400444477, 444000077, 444044477, 836666333, 866663333, 888803633, 888806333, 888880633, 888886333, 8888800033, 8888888033, 88888883333, 440444444477, 7777777777921, 8888888888333, 40000000000777, 44444444400077, 40444444444444477, 44444444444444477, 88888888888888633, 999999999999999121, 8<sub>77</sub>33|146|79|
|primes == 3 mod 4|3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, 200891, 208891, 228299, 282299, 545551, 608851, 686051, 822299, 828899, 848851, 866051, 880091, 885551, 888091, 888451, 902299, 909299, 909899, 2000291, 2888299, 2888891, 8000099, 8000891, 8000899, 8028299, 8808299, 8808551, 8880551, 8888851, 9000451, 9000899, 9908099, 9980099, 9990899, 9998099, 9999299, 60000851, 60008651, 60086651, 60866651, 68666651, 80088299, 80555551, 80888299, 88808099, 88808899, 88880899, 90000299, 90080099, 222222899, 800888899, 808802899, 808880099, 808888099, 888800299, 888822899, 992222299, 2222288899, 8808888899, 8888800099, 8888888299, 8888888891, 48555555551, 555555555551, 999999999899, 88888888888099, 2228888888888899, 9222222222222299, 2288888888888888888888899, 8<sub>43</sub>99, 86<sub>47</sub>51, 2<sub>19151</sub>99|113|19153|
|palindromic primes|2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889, 989040989, 996686699, 996989699, 999686999, 90689098609, 90899999809, 90999899909, 96099899069, 96600800669, 96609890669, 98000000089, 98844444889, 9009004009009, 9099094909909, 9600098900069, 9668000008669, 9699998999969, 9844444444489, 9899900099989, 9900004000099, 9900994990099, 900006898600009, 900904444409009, 966666989666669, 966668909866669, 966699989996669, 999090040090999, 999904444409999, 90000006860000009, 90000008480000009, 90000089998000009, 90999444444499909, 96000060806000069, 99900944444900999, 99990009490009999, 99999884448899999, 9000090994990900009, 9000094444444900009, 9666666080806666669, 9666666668666666669, 9909999994999999099, 9999444444444449999, 9999909994999099999, 9999990994990999999, 900000000080000000009, 900999994444499999009, 90000000009490000000009, 90909444444444444490909, 98999999444444499999989, 9904444444444444444444099, 999999999844444448999999999, 90944444444444444444444444909, 99999999999944444999999999999, 99999999999999499999999999999, 9999999999990004000999999999999, 900000000999999949999999000000009, 989999999999998444899999999999989, 9000000999999999994999999999990000009, ..., 994<sub>34019</sub>99, ... (this set is not known to be complete)|≥ 87|≥ 34023|
|composites|4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731|32|3|
|squares|1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876, 2707700770820007076, 7078287780880770276, 7808287827720727876, 8008002202002207876, 27282772777702807876, 70880800720008787876, 72887222220777087876, 80028077888770207876, 80880700827207270276, 87078270070088278276, 88002002000028027076, 2882278278888228807876, 8770777780888228887076, 77700027222828822007876, 702087807788807888287876, 788708087882007280808827876, 880070008077808877000002276, 888000227087070707880827076, 888077027227228277087787076, 888588886555505085888555556, 7770000800780088788282227776, 7782727788888878708800870276, 5000060065066660656065066555556, 8070008800822880080708800087876, 80787870808888808272077777227076, 800008088070820870870077778827876, 822822722220080888878078820887876, ... (this set is currently not known, and might be extremely difficult to found)|≥ 55|≥ 33|
|powers of 2|1, 2, 4, 8, 65536 (this set is conjectured to be complete by Jeffrey Shallit, not proven, however of course, if all powers of 2 except 65536 contain at least one of 1, 2, 4, 8, then this conjecture is true, only powers of 16 can be exceptions)|≥ 5|≥ 5|
|multiples of 3|0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 42, 45, 48, 51, 54, 57, 72, 75, 78, 81, 84, 87, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 411, 414, 417, 441, 444, 447, 471, 474, 477, 522, 525, 528, 552, 555, 558, 582, 585, 588, 711, 714, 717, 741, 744, 747, 771, 774, 777, 822, 825, 828, 852, 855, 858, 882, 885, 888|76|3|
|multiples of 4|0, 4, 8, 12, 16, 32, 36, 52, 56, 72, 76, 92, 96|13|2|
|range of Euler's totient function|1, 2, 4, 6, 8, 30, 70, 500, 900, 990, 5590, 9550, 555555555550|13|12|
|range of Dedekind psi function|1, 3, 4, 6, 8, 20, 72, 90, 222, 252, 500, 522, 552, 570, 592, 750, 770, 992, 7000, 5<sub>69</sub>0|20|70|
|range of "Euler's totient function + 1"|2, 3, 5, 7, 9, 11, 41, 61, 81|9|2|
|range of "Euler's totient function + 2"|3, 4, 6, 8, 10, 12, 20, 22, 50, 72, 90, 770, 992, 5592, 9552, 555555555552 (this set is conjectured to be complete, not proven, this conjecture is true if and only if there are no totients of the form 6{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*<sup>*n*</sup> with *p* prime and *n* odd)|16~17|12 or > 5000|
|range of "Euler's totient function + 3"|4, 5, 7, 9, 11, 13, 21, 23, 31, 33, 61, 63, 81, 83|14|2|
|range of "Euler's totient function + 4"|5, 6, 8, 10, 12, 14, 20, 22, 24, 32, 34, 40, 44, 70, 74, 92, 300, 472, 772, 900, 904, 994 (this set is conjectured to be complete, not proven, this conjecture is true if and only if there are no totients of the form {3,9}26 or {3,9}86, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*<sup>*n*</sup> with *p* prime and *n* odd)|≥ 22|3 or > 5000|
|range of "Euler's totient function + 5"|6, 7, 9, 11, 13, 15, 21, 23, 25, 33, 35, 41, 45, 51, 53, 83, 85, 301, 443, 505, 801, 881, 555555555555 (this set is conjectured to be complete, not proven, this conjecture is true if and only if there are no totients of the form 3{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (*p*−1)×*p*<sup>*n*</sup> with *p* prime and *n* odd)|23 or 24|12 or > 5000|

Tools about this research: (in fact, you can also use *Wolfram Alpha* (https://www.wolframalpha.com/) or online *Magma* calculator (http://magma.maths.usyd.edu.au/calc/) or *Pari*/*GP* (https://pari.math.u-bordeaux.fr/) or *Wolfram Mathematica* (https://www.wolfram.com/mathematica/) or *Maple* (https://www.maplesoft.com/))

Prime checkers:

1. https://t5k.org/curios/includes/primetest.php
2. https://www.numberempire.com/primenumbers.php
3. http://www.numbertheory.org/php/lucas.html
4. http://www.javascripter.net/faq/numberisprime.htm
5. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm
6. http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type *x* and click "prime?")
7. http://www.math.com/students/calculators/source/prime-number.htm
8. https://www.calculatorsoup.com/calculators/math/prime-number-calculator.php
9. https://onlinemathtools.com/test-prime-number
10. https://www.bigprimes.net/primalitytest
11. https://www.archimedes-lab.org/primOmatic.html
12. http://www.sonic.net/~undoc/java/PrimeCalc.html
13. http://www.primzahlen.de/primzahltests/testverfahren.htm (in German)
14. http://www.proftnj.com/calcprem.htm (in French) (use the box "Rechercher si un nombre est premier" and click "Rechercher")
15. http://www.positiveintegers.org/ (just enter the number)
16. https://numdic.com/ (just enter the number)
17. https://numbermatics.com/ (just enter the number)
18. https://metanumbers.com/ (just enter the number)
19. https://int.darkbyte.ru/ (just enter the number)
20. https://www.numbersaplenty.com/ (just enter the number)
21. https://t5k.org/nthprime/ (calculate the *n*th prime)
22. http://factordb.com/nextprime.php (calculate the next (probable) prime above *N*, in fact, links 2, 6, 10, 11, 12, 13 can also calculate the next prime above *N*, besides, links 2, 6 can also calculate the previous prime below *N*)

Integer factorizers:

1. https://www.numberempire.com/numberfactorizer.php
2. https://www.alpertron.com.ar/ECM.HTM
3. http://www.javascripter.net/math/calculators/primefactorscalculator.htm
4. https://web.archive.org/web/20230122202627/https://betaprojects.com/calculators/prime_factors.html
5. https://www.emathhelp.net/calculators/pre-algebra/prime-factorization-calculator/
6. http://www.numbertheory.org/php/factor.html
7. https://primefan.tripod.com/Factorer.html
8. https://www.calculatorsoup.com/calculators/math/prime-factors.php
9. https://www.calculator.net/prime-factorization-calculator.html
10. http://www.se16.info/js/factor.htm
11. http://math.fau.edu/Richman/mla/factor-f.htm
12. http://www.rsok.com/~jrm/factor.html
13. http://www.brennen.net/primes/FactorApplet.html (need run with Java)
14. https://web.archive.org/web/20161004191531/http://britton.disted.camosun.bc.ca/jbprimefactor.htm
15. http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html
16. http://www.analyzemath.com/Calculators_3/prime_factors.html
17. https://www.archimedes-lab.org/primOmatic.html
18. http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser")
19. http://www.positiveintegers.org/ (just enter the number)
20. https://numdic.com/ (just enter the number)
21. https://numbermatics.com/ (just enter the number)
22. https://metanumbers.com/ (just enter the number)
23. https://int.darkbyte.ru/ (just enter the number)
24. https://www.numbersaplenty.com/ (just enter the number)
25. http://factordb.com/ (online factor database)
26. https://578d0722p8.goho.co/index.html (more types of numbers in the online factor database)
27. http://myfactorcollection.mooo.com:8090/dbio.html (online factor database for numbers of the form *b*<sup>*n*</sup>±1)
28. https://web.archive.org/web/20120722020628/http://homes.cerias.purdue.edu/~ssw/cun/prime.php (online factor database for numbers of the form *b*<sup>*n*</sup>±1 for 2 ≤ *b* ≤ 12)
29. https://web.archive.org/web/20120330032919/http://homes.cerias.purdue.edu/~ssw/cun/clientold.html (online factor database for numbers of the form *b*<sup>*n*</sup>±1 for 2 ≤ *b* ≤ 12)

Base converters:

1. https://baseconvert.com/
2. https://baseconvert.com/high-precision
3. https://baseconvert.com/ieee-754-floating-point (for IEEE 754 (https://en.wikipedia.org/wiki/IEEE_754))
4. https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1
5. https://www.calculand.com/unit-converter/zahlen.php?og=Base62&ug=1
6. https://www.calculand.com/unit-converter/zahlen.php?og=Base64&ug=1
7. https://www.calculand.com/unit-converter/zahlen.php?og=Base85&ug=1
8. https://www.calculand.com/unit-converter/zahlen.php?og=System+calculand&ug=1
9. http://www.unitconversion.org/unit_converter/numbers.html
10. http://www.unitconversion.org/unit_converter/numbers-ex.html
11. http://www.math.com/students/converters/source/base.htm
12. https://www.dcode.fr/base-n-convert
13. https://www.cut-the-knot.org/binary.shtml
14. https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml
15. http://www.tonymarston.net/php-mysql/converter.php
16. http://math.fau.edu/Richman/mla/convert.htm
17. https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion
18. https://web.archive.org/web/20170204004954/http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip
19. http://www.kwuntung.net/hkunit/base/base.php (in Chinese)
20. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese)
21. http://www.positiveintegers.org/ (just enter the number)
22. https://numdic.com/ (just enter the number)
23. https://numbermatics.com/ (just enter the number)
24. https://metanumbers.com/ (just enter the number)
25. https://int.darkbyte.ru/ (just enter the number)
26. https://www.numbersaplenty.com/ (just enter the number)
27. http://factordb.com/index.php?showid=1000000000000000127 (you can change the "showid" to the *ID* for your number)

Expression generators:

1. https://stdkmd.net/nrr/exprgen.htm (only support base 10 forms)
2. https://www.numberempire.com/simplifyexpression.php (e.g. for the form 5{7} in base 11, type "5\*11^n+7\*(11^n-1)/10")

Lists of small primes: (see https://t5k.org/notes/faq/LongestList.html and https://t5k.org/notes/faq/x_digit_primes.html, although we can quickly determine whether a number < 10<sup>300</sup> is prime or not, but there is no list of all primes < 10<sup>300</sup>, since the space which we need is larger than the number of elementary particles (https://en.wikipedia.org/wiki/Elementary_particle) (i.e. quarks (https://en.wikipedia.org/wiki/Quark) and electrons (https://en.wikipedia.org/wiki/Electron)) in the universe (https://en.wikipedia.org/wiki/Universe))

1. https://t5k.org/lists/small/1000.txt
2. https://t5k.org/lists/small/10000.txt
3. https://t5k.org/lists/small/100000.txt
4. https://t5k.org/lists/small/millions/
5. https://oeis.org/A000040/b000040.txt
6. https://oeis.org/A000040/a000040.txt
7. https://oeis.org/A000040/b000040_1.txt
8. https://oeis.org/A000040/a000040_1B.7z
9. http://www.prime-numbers.org/
10. https://web.archive.org/web/20091027064420/http://geocities.com/primes_r_us/small/index.html
11. http://prime-numbers.org/sample.zip
12. https://metanumbers.com/prime-numbers
13. https://www.numberempire.com/primenumberstable.php
14. https://www.calculatorsoup.com/calculators/math/prime-numbers.php
15. https://www2.cs.arizona.edu/icon/oddsends/primes.htm
16. https://www.numbersaplenty.com/set/prime_number/more.php
17. https://cdn1.byjus.com/wp-content/uploads/2021/10/Prime-Numbers-from-1-to-1000.png
18. http://noe-education.org/D11102.php (in French)
19. https://web.archive.org/web/20060513054350/http://www.walter-fendt.de/m14i/primes_i.htm (in Italian)
20. https://primefan.tripod.com/500Primes1.html **(warning: this site incorrectly includes 1 as a prime and misses the primes 3229 and 3329)**
21. https://www.gutenberg.org/files/65/65.txt
22. http://www.primos.mat.br/indexen.html
23. https://www.walter-fendt.de/html5/men/primenumbers_en.htm
24. http://www.rsok.com/~jrm/printprimes.html
25. http://www.numbertheory.org/php/prime_generator.html
26. http://www.primzahlen.de/primzahltests/2-100003.htm (in German)
27. https://jocelyn.quizz.chat/np/cache/index.html (in French)
28. http://www.sosmath.com/tables/prime/prime.html
29. https://www.bigprimes.net/archive/prime
30. https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html
31. https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html
32. https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html
33. https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html (the longest list ever calculated, with all primes < 2<sup>64</sup> (but unlikely other lists here, the primes are not all stored))
34. https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers

Lists of factorizations of small integers:

1. http://primefan.tripod.com/500factored.html
2. http://www.sosmath.com/tables/factor/factor.html
3. https://sites.google.com/view/prime-factorization-of-integer
4. https://web.archive.org/web/20060210182347/http://bearnol.is-a-geek.com/Panfur%20Project/ **(warning: this site does not factor the composite numbers 15, 51, 85, 91, 255, 435, 451, 561, 595, 679, 703, 771, 1105, 1261, 1285, 1351, 1387, ...)**
5. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/
6. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/?infinity
7. https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of *N*)
8. http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of *N*)
9. http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of *N*)
10. https://en.wikipedia.org/wiki/Table_of_prime_factors
11. https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of *N*)
12. http://factordb.com/index.php?query=n&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (from *factordb*)

Lists of small integers in various bases:

1. https://en.wikipedia.org/wiki/Table_of_bases

Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php or https://download.mersenne.ca/) (some of these programs need to use *GMP* (https://gmplib.org/))

Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program):

1. *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64)
2. *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3)
3. *Primo* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309)
4. *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth)
5. *CHG* (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG)

Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) programs (https://www.rieselprime.de/ziki/Sieving_program):

1. *SRSieve* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve)
2. *MTSieve* (https://sourceforge.net/projects/mtsieve/, http://mersenneforum.org/rogue/mtsieve.html, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8)
3. *NewPGen* (https://t5k.org/programs/NewPGen/, https://t5k.org/bios/page.php?id=105, https://www.rieselprime.de/ziki/NewPGen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgenlinux)

Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program):

1. *GMP*-*ECM* (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM)
2. *MSieve* (https://sourceforge.net/projects/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64)
3. *GGNFS* (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS)
4. *CADO*-*NFS* (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0)
5. *YAFU* (http://bbuhrow.googlepages.com/home, https://github.com/bbuhrow/yafu)
6. *YTools* (https://github.com/bbuhrow/ytools)
7. *YSieve* (https://github.com/bbuhrow/ysieve)

For the files in this page:

* File "kernel *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings
* File "left *b*": *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that we were unable to determine if they contain a prime > *b* or not (i.e. *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that no prime member > *b* could be found, nor could the family be ruled out as only containing composites (only count the numbers > *b*)), these families are sorted by "the length *n* number in these families, from the smallest number to the largest number, this *n* is large enough such that *n* replaced to any larger number does not affect the sorting" (e.g. for base 17, we sort with B{0}B3 -> B{0}DB -> {B}2BE -> {B}2E -> {B}E9 -> {B}EE, since in this case 7 digits is enough, B0000B3 < B0000DB < BBBB2BE < BBBBB2E < BBBBBE9 < BBBBBEE, if the 7 replaced to any larger number, this result of the sorting will not change)
* File "special *b*": Non-linear families which cannot be ruled out by the "GMP.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc), but you can either handle them by hand or analyse them with the "famk.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/famk.cc)