Skip to content
Permalink
main
Switch branches/tags

Name already in use

A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Are you sure you want to create this branch?

minimal-elements-of-the-prime-numbers/README.md

Go to file
Update README.md
Latest commit 8eacc5c Nov 16, 2022 History
1 contributor

Users who have contributed to this file

514 lines (421 sloc) 56.9 KB
Raw Blame

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 delete the digit 4, to get the prime 19

Write down the prime 439 → I can delete the digit 9, to get the prime 43

Write down the prime 857 → I can delete zero digits, to get the prime 857

Write down the prime 2081 → I can delete the digit 0, to get the prime 281

Write down the largest known double Mersenne prime 170141183460469231731687303715884105727 (2127−1) → I can delete all digits except the third-leftmost 1 and the second-rightmost 3, to get the prime 13

Write down the largest known Fermat prime 65537 → I can delete the 6 and the 3, to get the prime 557 (also I can choose to delete the 6 and two 5's, to get the prime 37) (also I can choose to delete two 5's and the 3, to get the prime 67) (also I can choose to delete 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 delete 17 1's, to get the prime 11

Write down the prime 1000000000000000000000000000000000000000000000000000000000007 (which is the next prime after 1060) → I can delete all 0's, to get the prime 17

Write down the prime 95801 → I can delete the 9, to get the prime 5801

Write down the prime 946969 → I can delete the first 9 and two 6's, to get the prime 499

Write down the prime 90000000581 → I can delete five 0's, the 5, and the 8, to get the prime 9001

Write down the prime 8555555555555555555551 → I can delete the 8 and nine 5's, to get the prime 555555555551

Now we extend this prime game to bases 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 under a partial ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html) 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://primes.utm.edu/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html) 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://primes.utm.edu/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base) 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) with 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), 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, ..., 9 and A, B, ..., Z), using A−Z to represent digit values 10 to 35.

By the theorem that there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://primes.utm.edu/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 ordering, there must be only finitely such minimal elements in every base b.

This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf, https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf, https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf, https://doi.org/10.1080/10586458.2015.1064048, https://scholar.colorado.edu/downloads/hh63sw661, 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/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) with k-values < b, i.e. finding the smallest prime of the form k×bn+1 and k×bn−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, 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, 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 bn+k and bn−k (which are the dual forms of k×bn+1 and k×bn−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 bn+2, bn−2, bn+(b−1), bn−(b−1), 2×bn+1, 2×bn−1, (b−1)×bn+1, (b−1)×bn−1, for the same base 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, http://www.noprimeleftbehind.net/crus/vstats/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats/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. 25×3034205−1 is not minimal prime in base 30 in original definition, since it is OT34205 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.

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).

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://primes.utm.edu/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.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 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), 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://primes.utm.edu/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, 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.

This problem covers finding the smallest prime of these forms in the same base b (or proving that such prime does not exist): (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) b)

(bn−1)/(b−1) with n ≥ 2 (see http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://archive.ph/tf7jx, http://www.primenumbers.net/Henri/us/MersFermus.htm, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A065854, https://oeis.org/A279068, https://oeis.org/A128164, https://oeis.org/A285642)

bn+1 with n ≥ 1 (see 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, https://oeis.org/A228101, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A123669)

(bn+1)/2 (for odd b) with n ≥ 2 (see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)

2×bn+1 with n ≥ 1 (see https://mersenneforum.org/showthread.php?t=6918, https://mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624, https://oeis.org/A253178, https://oeis.org/A098872)

2×bn−1 with n ≥ 1 (see 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)

bn+2 with n ≥ 1 (see https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)

bn−2 with n ≥ 2 (see https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714, https://oeis.org/A292201)

(b−1)×bn+1 with n ≥ 1 (see https://www.rieselprime.de/ziki/Williams_prime_MP_least, https://www.rieselprime.de/ziki/Williams_prime_MP_table, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/477, https://oeis.org/A305531, https://oeis.org/A087139)

(b−1)×bn−1 with n ≥ 1 (see 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://sites.google.com/view/williams-primes, http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf, https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf, http://www.bitman.name/math/table/484, https://oeis.org/A122396)

bn+(b−1) with n ≥ 1 (see https://sites.google.com/view/williams-primes, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179)

bn−(b−1) with n ≥ 2 (see https://sites.google.com/view/williams-primes, https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435, https://oeis.org/A113516, https://oeis.org/A343589)

k×bn+1 for all k ≤ 12 with n ≥ 1 (see https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)

k×bn−1 for all k ≤ 12 with n ≥ 1 (see https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)

(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/primecount.txt, 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)

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)

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, https://en.wikipedia.org/wiki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html), algebraic factorization (https://en.wikipedia.org/wiki/Factorization_of_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html), or combine of them, 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)

b family why this family contain no primes > b
10 2{0}1 always divisible by 3
10 2{0}7 always divisible by 3
10 5{0}1 always divisible by 3
10 5{0}7 always divisible by 3
10 8{0}1 always divisible by 3
10 8{0}7 always divisible by 3
10 28{0}7 always divisible by 7
10 4{6}9 always divisible by 7
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 (non-simple family)
10 {0,7} always divisible by 7 (non-simple 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 only containing digits which are divisible by some d > 1 always divisible by d
3 1{0}1 always divisible by 2
4 2{0}1 always divisible by 3
5 11{0}3 always divisible by 3
5 3{0}11 always divisible by 3
6 4{0}1 always divisible by 5
7 1{0}1{0}1 always divisible by 3 (non-simple family)
7 1{0}3{0}5 always divisible by 3 (non-simple family)
7 1{0}5{0}3 always divisible by 3 (non-simple family)
7 3{0}1{0}5 always divisible by 3 (non-simple family)
7 3{0}5{0}1 always divisible by 3 (non-simple family)
7 5{0}1{0}3 always divisible by 3 (non-simple family)
7 5{0}3{0}1 always divisible by 3 (non-simple family)
8 2{0}5 always divisible by 7
8 4{0}3 always divisible by 7
8 6{0}1 always divisible by 7
8 44{0}3 always divisible by 3
8 6{0}11 always divisible by 3
9 {7}62 always divisible by 7
12 A{0}21 always divisible by 5
13 C{A}5 always divisible by 7
14 40{4}9 always divisible by 61
15 9{6}8 always divisible by 11
16 2{C}3 always divisible by 7
21 B0{H}6H always divisible by 4637
9 {1}5 always divisible by some element of {2,5}
9 2{7} always divisible by some element of {2,5}
9 {3}8 always divisible by some element of {2,5}
9 5{1} always divisible by some element of {2,5}
9 5{7} always divisible by some element of {2,5}
9 {7}2 always divisible by some element of {2,5}
9 {7}5 always divisible by some element of {2,5}
9 {1}6{1} always divisible by some element of {2,5} (non-simple family)
9 {3}{0}5 always divisible by some element of {2,5} (non-simple family)
11 2{5} always divisible by some element of {2,3}
11 3{5} always divisible by some element of {2,3}
11 3{7} always divisible by some element of {2,3}
11 4{7} always divisible by some element of {2,3}
11 {5}2 always divisible by some element of {2,3}
11 {5}3 always divisible by some element of {2,3}
11 {7}3 always divisible by some element of {2,3}
11 {7}4 always divisible by some element of {2,3}
14 4{0}1 always divisible by some element of {3,5}
14 B{0}1 always divisible by some element of {3,5}
14 3{D} always divisible by some element of {3,5}
14 A{D} always divisible by some element of {3,5}
14 1{0}B always divisible by some element of {3,5}
14 {D}3 always divisible by some element of {3,5}
14 {4}9 always divisible by some element of {3,5}
14 {8}5 always divisible by some element of {3,5}
8 6{4}7 always divisible by some element of {3,5,13} (special example, as the numbers with length ≥ 222 in this family contain "prime > b" subsequence, this prime is 42207)
13 3{0}95 always divisible by some element of {5,7,17}
13 95{0}3 always divisible by some element of {5,7,17}
16 {4}D always divisible by some element of {3,7,13}
16 {8}F always divisible by some element of {3,7,13}
17 7F{0}D always divisible by some element of {3,5,29}
17 D{0}7F always divisible by some element of {3,5,29}
20 8{0}1 always divisible by some element of {3,7}
20 D{0}1 always divisible by some element of {3,7}
20 7{J} always divisible by some element of {3,7}
20 C{J} always divisible by some element of {3,7}
20 1{0}D always divisible by some element of {3,7}
20 {J}7 always divisible by some element of {3,7}
21 {7}D always divisible by some element of {2,13,17}
23 7L{0}1 always divisible by some element of {3,5,53}
23 1{0}7L always divisible by some element of {3,5,53}
27 JP{0}1 always divisible by some element of {5,7,73}
27 1{0}JP always divisible by some element of {5,7,73}
30 A{0}9J always divisible by some element of {7,13,19,31}
32 A{0}1 always divisible by some element of {3,11}
32 N{0}1 always divisible by some element of {3,11}
32 9{V} always divisible by some element of {3,11}
32 M{V} always divisible by some element of {3,11}
32 1{0}N always divisible by some element of {3,11}
32 {V}9 always divisible by some element of {3,11}
32 8{0}V always divisible by some element of {3,5,41}
34 6{0}1 always divisible by some element of {5,7}
34 5{X} always divisible by some element of {5,7}
34 S{X} always divisible by some element of {5,7}
34 {X}5 always divisible by some element of {5,7}
9 {1} difference-of-squares factorization
8 1{0}1 sum-of-cubes factorization
9 3{1} difference-of-squares factorization
9 3{8} difference-of-squares factorization
9 {8}5 difference-of-squares factorization
9 3{8}35 difference-of-squares factorization
16 8{F} difference-of-squares factorization
16 {F}7 difference-of-squares factorization
16 {4}1 difference-of-squares factorization
16 B{4}1 difference-of-squares factorization
16 1{5} difference-of-squares factorization
16 8{5} difference-of-squares factorization
16 10{5} difference-of-squares factorization
16 A1{5} difference-of-squares factorization
16 7{3} difference-of-squares factorization
16 3{F}AF difference-of-squares factorization
16 30{F}AF difference-of-squares factorization
16 3{F}A0F difference-of-squares factorization
16 30{F}A0F difference-of-squares factorization
16 {5}45 difference-of-squares factorization
16 {C}B difference-of-squares factorization
16 {C}D Aurifeuillian factorization of x4+4×y4
16 {C}DD Aurifeuillian factorization of x4+4×y4
25 {1} difference-of-squares factorization
25 2{1} difference-of-squares factorization
25 5{1} difference-of-squares factorization
25 7{1} difference-of-squares factorization
25 C{1} difference-of-squares factorization
25 F{1} difference-of-squares factorization
25 M{1} difference-of-squares factorization
25 1F{1} difference-of-squares factorization
25 1{3} difference-of-squares factorization
25 1{8} difference-of-squares factorization
25 5{8} difference-of-squares factorization
25 A{3} difference-of-squares factorization
25 L{8} difference-of-squares factorization
25 {3}2 difference-of-squares factorization
25 {8}3 difference-of-squares factorization
25 {8}7 difference-of-squares factorization
27 8{0}1 sum-of-cubes factorization
27 1{0}8 sum-of-cubes factorization
27 {D}E sum-of-cubes factorization
27 7{Q} difference-of-cubes factorization
27 {Q}J difference-of-cubes factorization
27 9{G} difference-of-cubes factorization
32 1{0}1 sum-of-5th-powers factorization
32 {1} difference-of-5th-powers factorization
36 3{7} difference-of-squares factorization
36 3{Z} difference-of-squares factorization
36 8{Z} difference-of-squares factorization
36 O{Z} difference-of-squares factorization
36 {Z}B difference-of-squares factorization
36 8{Z}B difference-of-squares factorization
36 F{Z}B difference-of-squares factorization
36 O{5} difference-of-squares factorization
36 O{7} difference-of-squares factorization
36 {9}1 difference-of-squares factorization
36 T{9}1 difference-of-squares factorization
36 {S}J difference-of-squares factorization
14 8{D} combine of factor 5 and difference-of-squares factorization
12 {B}9B combine of factor 13 and difference-of-squares factorization
14 {D}5 combine of factor 5 and difference-of-squares factorization
17 1{9} combine of factor 2 and difference-of-squares factorization
17 7{9} combine of factor 2 and difference-of-squares factorization
17 {9}2 combine of factor 2 and difference-of-squares factorization
17 {9}8 combine of factor 2 and difference-of-squares factorization
19 1{6} combine of factor 5 and difference-of-squares factorization
19 {6}5 combine of factor 5 and difference-of-squares factorization
19 7{2} combine of factor 5 and difference-of-squares factorization
19 89{6} combine of factor 5 and difference-of-squares factorization
24 3{N} combine of factor 5 and difference-of-squares factorization
24 5{N} combine of factor 5 and difference-of-squares factorization
24 8{N} combine of factor 5 and difference-of-squares factorization
24 {6}1 combine of factor 5 and difference-of-squares factorization
24 {N}LN combine of factor 5 and difference-of-squares factorization
33 F{W} combine of factor 17 and difference-of-squares factorization
33 {W}H combine of factor 17 and difference-of-squares factorization
34 1{B} combine of factor 5 and difference-of-squares factorization
34 8{X} combine of factor 5 and difference-of-squares factorization
34 {X}P combine of factor 5 and difference-of-squares factorization

Also families which contain only one very small prime > b:

b family why this family contains only one prime > b
4 {1} difference-of-squares factorization, but 11 is prime, and 11 is the only prime > b in this family
8 {1} difference-of-cubes factorization, but 111 is prime, and 111 is the only prime > b in this family
16 {1} difference-of-squares factorization, but 11 is prime, and 11 is the only prime > b in this family
27 {1} difference-of-cubes factorization, but 111 is prime, and 111 is the only prime > b in this family
27 {G}7 difference-of-cubes factorization, but G7 is prime, and G7 is the only prime > b in this family
36 {1} difference-of-squares factorization, but 11 is prime, and 11 is the only prime > b in this family

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 50000 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 the base 23 family 9{E} is 9E800873 which when written in decimal contains 1090573 digits (technically, probable primality tests were used to show this (which have a very small chance of making an error (https://primes.utm.edu/notes/prp_prob.html, https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.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) run far too slowly to run on a number of this size unless either N−1 (https://primes.utm.edu/prove/prove3_1.html) or N+1 (https://primes.utm.edu/prove/prove3_2.html) (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://primes.utm.edu/glossary/xpage/Cyclotomy.html) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html), see the article http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf for the case that either N−1 or N+1 (or both) can be ≥ 1/3 factored, 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://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/CHG.GP.txt, https://primes.utm.edu/bios/page.php?id=797) to prove its primality (see https://mersenneforum.org/showpost.php?p=528149&postcount=3), for the examples of the numbers which are proven prime by CHG, see https://primes.utm.edu/primes/page.php?id=126454, https://primes.utm.edu/primes/page.php?id=131964, https://primes.utm.edu/primes/page.php?id=123456, https://primes.utm.edu/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"), however, factordb (http://factordb.com/) lacks the ability to verify CHG proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165)

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×bn+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, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin primality test (https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://primes.utm.edu/glossary/xpage/MillersTest.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html) 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, it is possible to use a sieving process (https://www.rieselprime.de/ziki/Sieving, https://mathworld.wolfram.com/Sieve.html) to find divisors rather than using trial division (https://en.wikipedia.org/wiki/Trial_division, https://primes.utm.edu/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html).

To do this, we made use of Geoffrey Reynolds’ SRSIEVE software (https://www.bc-team.org/app.php/dlext/?cat=3, https://mersenneforum.org/attachment.php?attachmentid=16377&d=1499103807, https://archive.ph/XrJkw, http://www.rieselprime.de/dl/CRUS_pack.zip, https://primes.utm.edu/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, for the README see https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/srsieve%20README.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/srsieve%20README2.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr1sieve%20README.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr1sieve%20README2.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr1sieve%20README3.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr2sieve%20README.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr2sieve%20README2.txt, https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/sr2sieve%20README3.txt). This program uses the baby-step giant-step algorithm to find all primes p which divide a×bn+c where p and n lie in a specified range (also, this program was updated so that it also removes the n such that a×bn+c has algebraic factors (e.g. difference-of-squares factorization, sum/difference-of-cubes factorization, Aurifeuillian factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html) of x4+4×y4), see https://mersenneforum.org/showpost.php?p=452132&postcount=66 and https://mersenneforum.org/showthread.php?t=21916). Since this program cannot handle the general case (a×bn+c)/gcd(a+c,b−1) when gcd(a+c,b−1) > 1 we only used it to sieve the sequence a×bn+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×bn+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×bn+c, but 2 may not divide (a×bn+c)/gcd(a+c,b−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960).

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 Penn´e (http://jpenne.free.fr/index2.html, https://primes.utm.edu/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, for the README see https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/llr%20README.txt) or PFGW (https://sourceforge.net/projects/openpfgw/, https://primes.utm.edu/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, for the README see https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/pfgw%20README.txt). Although undocumented, it is possible to run these two programs on numbers of the form (a×bn+c)/gcd(a+c,b−1) when gcd(a+c,b−1) > 1, 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 < 1025000 for the solved or near-solved bases (bases b with ≤ 3 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, 28, 30), we employed PRIMO by Marcel Martin (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://primes.utm.edu/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo), an elliptic curve primality proving (https://primes.utm.edu/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://primes.utm.edu/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html) implementation.

We have completely solved this problem for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 (thus, currently we can complete the classification of the minimal primes in these bases), 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://primes.utm.edu/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html) > 1025000 in place of proven primes, besides, we have completely solved this problem for bases b = 13, 17, 19, 21, 23, 26, 28, 36 (if we allow strong probable primes in place of proven primes) except the families listed in the "left b" files (see the condensed table below for the searching limit of these families).

We are unable to determine if these families contain a prime (only count the numbers > base (b)) or not, i.e. these families have no known prime members, nor can they be ruled out as only containing composites, and all of these families are excepted to contain primes.

For base 17, the smallest prime in family {B}2BE may or may not be minimal prime, since another unsolved family is {B}2E.

For base 19, the smallest prime in family {2}7A may or may not be minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be minimal prime, since another unsolved family is FH0{H}.

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be minimal prime, since another unsolved family is H{D}.

There are also unproven probable primes (however, in this project our results assume that they are in fact primes, since they are > 1025000 and the probability that they are in fact composite is < 10−2000, see https://primes.utm.edu/notes/prp_prob.html), 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, 30 are:

b index of this minimal prime in base b (assuming the primality of all probable primes in base b) base-b form of the unproven probable prime algebraic ((a×bn+c)/gcd(a+c,b−1)) form of the unproven probable prime
11 1068 5762668
13 3194 C523755C
13 3195 8032017111
13 3196 95197420
16 2345 DB32234
16 2346 472785DD
16 2347 3116137AF
22 8003 BK220015
28 25526 N624051LR
28 25527 5OA31238F
28 25528 O4O945359
30 2618 I024608D

All these numbers are strong probable primes (https://en.wikipedia.org/wiki/Strong_pseudoprime, https://primes.utm.edu/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 strong Lucas probable primes (https://en.wikipedia.org/wiki/Lucas_pseudoprime#Strong_Lucas_pseudoprimes, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (P, Q) defined by Selfridge's Method A (see https://oeis.org/A217255), and trial factored to 1016 (thus, all these numbers are Baillie–PSW probable primes.

Primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://primes.utm.edu/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html) for large proven primes (> 10300) for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 28, 30:

b index of this minimal prime in base b base-b form of the minimal prime algebraic ((a×bn+c)/gcd(a+c,b−1)) form of the minimal prime primality certificate for the minimal prime
9 149 763292 (31×9330−19)/4 http://factordb.com/cert.php?id=1100000002359003642
9 150 2768607 (23×9688−511)/8 http://factordb.com/cert.php?id=1100000002495467486
9 151 30115811 3×91160+10 http://factordb.com/cert.php?id=1100000002376318423
11 1065 A71358 11715−58 http://factordb.com/cert.php?id=1100000003576826487
11 1066 775944 (7×11761−367)/10 http://factordb.com/cert.php?id=1100000002505568840
11 1067 5571011 (607×111011−7)/10 http://factordb.com/cert.php?id=1100000002361376522
13 3184 9968B (3×13969+5)/4 http://factordb.com/cert.php?id=1100000000258566244
13 3185 101295181 131298+274 http://factordb.com/cert.php?id=1100000002615445013
13 3186 913625 (3×131363−19)/4 http://factordb.com/cert.php?id=1100000002321017776
13 3187 715041 (7×131505−79)/12 http://factordb.com/cert.php?id=1100000002320890755
13 3188 93015511 120×131552+1 proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html), since N−1 is trivially fully factored
13 3189 72022972 93×132298+2 http://factordb.com/cert.php?id=1100000002632396910
13 3190 1770270317 267×132705+20 http://factordb.com/cert.php?id=1100000003590430825
13 3191 39062661 48×136267+1 proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html), since N−1 is trivially fully factored
13 3192 B06540BBA 11×136543+2012 http://factordb.com/cert.php?id=1100000002616382906
13 3193 C1063192 1310633−50 http://factordb.com/cert.php?id=1100000003590493750
14 650 4D19698 5×1419698−1 proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html), since N+1 is trivially fully factored
16 2337 D91052 (68×161052−3)/5 http://factordb.com/cert.php?id=1100000002321036020
16 2338 FAF106245 251×161064−187 http://factordb.com/cert.php?id=1100000003588387610
16 2339 F81517F (233×161518+97)/15 http://factordb.com/cert.php?id=1100000000633744824
16 2340 201713321 2×161716+801 http://factordb.com/cert.php?id=1100000003588386735
16 2341 300F1960AF 769×161962−81 http://factordb.com/cert.php?id=1100000003588368750
16 2342 90354291 9×163544+145 http://factordb.com/cert.php?id=1100000000633424191
16 2343 5BC3700D (459×163701+1)/5 http://factordb.com/cert.php?id=1100000000993764322
16 2344 D0B17804 (3131×1617804−11)/15 http://factordb.com/cert.php?id=1100000003589278511
18 547 80298B 8×18299+11 http://factordb.com/cert.php?id=1100000002355574745
18 548 H766FH 18768−37 http://factordb.com/cert.php?id=1100000003590430490
18 549 C06268C5 12×186270+221 http://factordb.com/cert.php?id=1100000003590442437
20 3312 501163AJ 5×201165+219 http://factordb.com/cert.php?id=1100000003590502412
20 3313 CD2449 (241×202449−13)/19 http://factordb.com/cert.php?id=1100000002325393915
20 3314 G06269D 16×206270+13 http://factordb.com/cert.php?id=1100000003590539457
22 7998 K0760EC1 20×22763+7041 http://factordb.com/cert.php?id=1100000000632724415
22 7999 J0767IGGJ 19×22771+199779 http://factordb.com/cert.php?id=1100000003591362567
22 8000 7959K7 (22961+857)/3 http://factordb.com/cert.php?id=1100000003591361817
22 8001 L2385KE7 222388−653 http://factordb.com/cert.php?id=1100000003591360774
22 8002 738152L (223817−289)/3 http://factordb.com/cert.php?id=1100000003591359839
24 3405 N2644LLN 242647−1201 http://factordb.com/cert.php?id=1100000003593270089
24 3406 D2698LD (13×242700+4403)/23 http://factordb.com/cert.php?id=1100000003593269876
24 3407 A029518ID 10×242954+5053 http://factordb.com/cert.php?id=1100000003593269654
24 3408 88N5951 201×245951−1 proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html), since N+1 is trivially fully factored
24 3409 N00N8129LN 13249×248131−49 http://factordb.com/cert.php?id=1100000003593391606
28 25513 70748M5 7×28750+621 http://factordb.com/cert.php?id=1100000003850161956
28 25514 4A0804B 122×28805+11 http://factordb.com/cert.php?id=1100000003850161957
28 25515 LK925F (587×28926−155)/27 http://factordb.com/cert.php?id=1100000000840839978
28 25516 J01071AC5 19×281074+8181 http://factordb.com/cert.php?id=1100000003850161959
28 25517 J01252J5 19×281254+537 http://factordb.com/cert.php?id=1100000003850161963
28 25518 513046F (5×281306+1021)/27 http://factordb.com/cert.php?id=1100000003850161964
28 25519 51332P8P (5×281335+426163)/27 http://factordb.com/cert.php?id=1100000003850161965
28 25520 5I1370F (17×281371−11)/3 http://factordb.com/cert.php?id=1100000003850161972
28 25521 A14236F (10×281425−2899)/27 http://factordb.com/cert.php?id=1100000000840839947
28 25522 G01899AN 16×281901+303 http://factordb.com/cert.php?id=1100000003850161973
28 25523 537468P (5×283748+2803)/27 http://factordb.com/cert.php?id=1100000003850161974
28 25524 QO423969 (242×284241−4679)/9 http://factordb.com/cert.php?id=1100000000840839934
28 25525 D0526777D 13×285270+5697 http://factordb.com/cert.php?id=1100000003850151420
30 2616 C010221 12×301023+1 proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html), since N−1 is trivially fully factored
30 2617 54882J (5×304883+401)/29 http://factordb.com/cert.php?id=1100000002327649423
30 2619 OT34205 25×3034205−1 proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html), since N+1 is trivially fully factored

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)

b number of minimal primes base b base-b form of the top 5 known minimal prime base b length of the top 5 known minimal prime base b algebraic ((a×bn+c)/gcd(a+c,b−1)) form of the top 5 known minimal prime base b 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 3 0 ---
3 3 111
21
12		 3
2
2		 13
7
5		 0 ---
4 5 221
31
23
13
11		 3
2
2
2
2		 41
13
11
7
5		 0 ---
5 22 109313
300031
44441
33331
33001		 96
6
5
5
5		 595+8
9391
3121
2341
2251		 0 ---
6 11 40041
4441
4401
51
45		 5
4
4
2
2		 5209
1033
1009
31
29		 0 ---
7 71 3161
51071
3601
1100021
531101		 17
10
8
7
6		 (717−5)/2
36×78+1
(78−47)/2
134471
91631		 0 ---
8 75 42207
51325
7121
7777461
7471		 221
15
13
11
9		 (4×8221+17)/7
(5×815−173)/7
813−7
(28669×87−25)/7
(53×88−25)/7		 0 ---
9 151 30115811
2768607
763292
56136
102557		 1161
689
331
38
28		 3×91160+10
(23×9688−511)/8
(31×9330−19)/4
(409×936−1)/8
927+52		 0 ---
10 77 502827
5111
80555551
66600049
66000049		 31
12
8
8
8		 5×1030+27
(5×1012−41)/9
80555551
66600049
66000049		 0 ---
11 1068 5762668
5571011
775944
A71358
8522005		 62669
1013
761
715
223		 (57×1162668−7)/10
(607×111011−7)/10
(7×11761−367)/10
11715−58
(17×11222−111)/2		 0 ---
12 106 403977
B0279B
B699B
AA000001
B00099B		 42
30
9
8
7		 4×1241+91
11×1229+119
129−313
388177921
32847239		 0 ---
13 3196~3197 95197420
8032017111
C523755C
C1063192
B06540BBA		 197421
32021
23757
10633
6544		 (113×13197420−5)/12
8×1332020+183
(149×1323756+79)/12
1310633−50
11×136543+2012		 1 200000
14 650 4D19698
34D708
8D14185
886B
408349		 19699
710
144
87
86		 5×1419698−1
47×14708−1
9×14143−79
(8×1487+31)/13
4×1485+65		 0 ---
15 1284 715597
E145397
9610408
773CE
759CCE		 157
148
107
75
62		 (15157+59)/2
15148−2558
(66×15106−619)/7
(1575+163)/2
(1562+2413)/2		 0 ---
16 2347 3116137AF
472785DD
DB32234
D0B17804
5BC3700D		 116139
72787
32235
17806
3703		 (16116139+619)/5
(4×1672787+2291)/15
(206×1632234−11)/15
(3131×1617804−11)/15
(459×163701+1)/5		 0 ---
17 10408~10428 570513101
E9B44732
D0GD37096
G732072F
15024325D		 51313
44734
37099
32074
24328		 92×1751311+1
(3963×1744732−11)/16
(60381×1737096−13)/16
(263×1732073+121)/16
22×1724326+13		 20 53000
18 549 C06268C5
H766FH
80298B
C0116F5
HD93		 6271
768
300
119
94		 12×186270+221
18768−37
8×18299+11
12×18118+275
(302×1893−13)/17		 0 ---
19 31400~31435 D17D0197501
F10185231
5D171075
H1709102
E016412111		 19755
18526
17109
17093
16416		 89674×1919751+1
286×1918524+1
(103×1917108−157)/18
(17×1917093−6101)/18
14×1916415+381		 35 20000
20 3314 G06269D
CD2449
501163AJ
J65505J
JCJ629		 6271
2450
1166
658
631		 16×206270+13
(241×202449−13)/19
5×201165+219
20658−7881
393×20629−1		 0 ---
21 13373~13395 5D0198481 19851 118×2119849+1 22 20000
22 8003 BK220015 22003 (251×2222002−335)/21 0 ---
23 65144~65276 71906733 19069 (7×2319069−2119)/22 132 20000
24 3409 N00N8129LN 8134 13249×248131−49 0 ---
25
26 25253~25259 6K233005 23302 (34×2623301−79)/5 6 38000
27
28 25528~25529 O4O945359 94538 (6092×2894536−143)/9 1 543202
29
30 2619 OT34205 34206 25×3034205−1 0 ---
31
32
33
34
35
36 35284~35290 7K26567Z 26569 (53×3626568+101)/7 6 36000

Related links:

https://primes.utm.edu/primes/lists/all.txt (top definitely primes)

https://primes.utm.edu/primes/lists/short.txt (definitely primes with ≥ 800000 decimal digits)

https://primes.utm.edu/primes/download.php (index page of top definitely primes)

https://primes.utm.edu/primes/search.php (search page of top definitely primes)

https://primes.utm.edu/primes/search.php?Advanced=1 (advanced search page of top definitely primes)

https://primes.utm.edu/primes/search_proth.php (search page of top definitely primes of the form k×bn±1)

http://www.primenumbers.net/prptop/prptop.php (top probable primes)

http://www.primenumbers.net/prptop/searchform.php (search page of top probable primes)

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/))

Prime checkers:

  1. https://primes.utm.edu/curios/includes/primetest.php
  2. https://www.numberempire.com/primenumbers.php
  3. http://www.numbertheory.org/php/lucas.html
  4. http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type x and click "prime?")
  5. https://www.bigprimes.net/primalitytest
  6. http://www.proftnj.com/calcprem.htm
  7. https://www.archimedes-lab.org/primOmatic.html
  8. http://www.sonic.net/~undoc/java/PrimeCalc.html

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://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. http://www.se16.info/js/factor.htm
  9. http://math.fau.edu/Richman/mla/factor-f.htm

Base converters:

  1. https://baseconvert.com/
  2. https://www.calculand.com/unit-converter/zahlen.php
  3. https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml
  4. http://www.tonymarston.net/php-mysql/converter.html
  5. http://www.kwuntung.net/hkunit/base/base.php (in Chinese)
  6. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese)

List of small primes:

  1. https://primes.utm.edu/lists/small/1000.txt
  2. https://primes.utm.edu/lists/small/10000.txt
  3. https://primes.utm.edu/lists/small/100000.txt
  4. https://primes.utm.edu/lists/small/millions/
  5. https://oeis.org/A000040/a000040.txt
  6. https://oeis.org/A000040/b000040_1.txt
  7. https://oeis.org/A000040/a000040_1B.7z
  8. https://metanumbers.com/prime-numbers
  9. https://www2.cs.arizona.edu/icon/oddsends/primes.htm
  10. http://noe-education.org/D11102.php (in French)
  11. https://archive.ph/dFHCI (in Italian)
  12. https://primefan.tripod.com/500Primes1.html
  13. https://www.gutenberg.org/files/65/65.txt
  14. http://www.primos.mat.br/indexen.html
  15. https://www.walter-fendt.de/html5/men/primenumbers_en.htm
  16. http://www.rsok.com/~jrm/printprimes.html
  17. https://jocelyn.quizz.chat/np/cache/index.html
  18. 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://en.wikipedia.org/wiki/Table_of_prime_factors

Lists of small integers in various bases:

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

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 sorting will not change)

See my article about this research: https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub