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 (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/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*×*b*^*n*+1 and *k*×*b*^*n*−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, 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*^*n*+*k* and *b*^*n*−*k* (which are the dual forms of *k*×*b*^*n*+1 and *k*×*b*^*n*−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*^*n*+2, *b*^*n*−2, *b*^*n*+(*b*−1), *b*^*n*−(*b*−1), 2×*b*^*n*+1, 2×*b*^*n*−1, (*b*−1)×*b*^*n*+1, (*b*−1)×*b*^*n*−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 (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 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*) |family|smallest allowed *n*|*OEIS* sequences for the smallest *n* such that this form is prime|references |:---|:---|:---|:---| |(*b*^*n*−1)/(*b*−1)|2|https://oeis.org/A084740
https://oeis.org/A084738 (corresponding primes)
https://oeis.org/A065854 (prime *b*)
https://oeis.org/A279068 (prime *b*, corresponding primes)
https://oeis.org/A128164 (*n* = 2 not allowed)
https://oeis.org/A285642 (*n* = 2 not allowed, corresponding primes)|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 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf)| |*b*^*n*+1|1|https://oeis.org/A228101 (*log*₂ of *n*)
https://oeis.org/A079706
https://oeis.org/A084712 (corresponding primes)
https://oeis.org/A123669 (*n* = 1 not allowed, corresponding primes)|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| |(*b*^*n*+1)/2 (for odd *b*)|2||http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt| |2×*b*^*n*+1|1|https://oeis.org/A119624
https://oeis.org/A253178 (only bases which have possible primes)
https://oeis.org/A098872 (*b* divisible by 6)|https://mersenneforum.org/showthread.php?t=6918
https://mersenneforum.org/showthread.php?t=19725| |2×*b*^*n*−1|1|https://oeis.org/A119591
https://oeis.org/A098873 (*b* divisible by 6)|https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217| |*b*^*n*+2|1|https://oeis.org/A138066
https://oeis.org/A084713 (corresponding primes)
https://oeis.org/A138067 (*n* = 1 not allowed)| |*b*^*n*−2|2|https://oeis.org/A250200
https://oeis.org/A255707 (*n* = 1 allowed)
https://oeis.org/A084714 (*n* = 1 allowed, corresponding primes)
https://oeis.org/A292201 (prime *b*, *n* = 1 allowed)|https://www.primepuzzles.net/puzzles/puzz_887.htm (*n* = 1 allowed)| |3×*b*^*n*+1|1|https://oeis.org/A098877 (*b* divisible by 6)|| |3×*b*^*n*−1|1|https://oeis.org/A098876 (*b* divisible by 6)|| |2×*b*^*n*+3|1||https://www.primegrid.com/forum_thread.php?id=9538| |*b*^*n*/2+1 (for even *b*)|2||https://www.primegrid.com/forum_thread.php?id=9538| |(*b*−1)×*b*^*n*+1|1|https://oeis.org/A305531
https://oeis.org/A087139 (prime *b*, *n* replaced by *n*+1)|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| |(*b*−1)×*b*^*n*−1|1|https://oeis.org/A122396 (prime *b*, *n* replaced by *n*+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://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| |*b*^*n*+(*b*−1)|1|https://oeis.org/A076845
https://oeis.org/A076846 (corresponding primes)
https://oeis.org/A078178 (*n* = 1 not allowed)
https://oeis.org/A078179 (*n* = 1 not allowed, corresponding primes)|https://sites.google.com/view/williams-primes| |*b*^*n*−(*b*−1)|2|https://oeis.org/A113516
https://oeis.org/A343589 (corresponding primes)|https://sites.google.com/view/williams-primes
https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime *b*)
http://www.bitman.name/math/table/435 (prime *b*)| |*k*×*b*^*n*+1 for all 2 ≤ *k* ≤ 12|1||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://mersenneforum.org/showthread.php?t=10354| |*k*×*b*^*n*−1 for all 2 ≤ *k* ≤ 12|1||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) (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) |*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}
divisible by 2 if the length is even, divisible by 5 if the length is odd| |9|2{7}|always divisible by some element of {2,5}
divisible by 2 if the length is odd, divisible by 5 if the length is even| |9|{3}8|always divisible by some element of {2,5}
divisible by 2 if the length is odd, divisible by 5 if the length is even| |9|5{1}|always divisible by some element of {2,5}
divisible by 2 if the length is even, divisible by 5 if the length is odd| |9|5{7}|always divisible by some element of {2,5}
divisible by 2 if the length is even, divisible by 5 if the length is odd| |9|{7}2|always divisible by some element of {2,5}
divisible by 2 if the length is odd, divisible by 5 if the length is even| |9|{7}5|always divisible by some element of {2,5}
divisible by 2 if the length is even, divisible by 5 if the length is odd| |9|{1}6{1}|always divisible by some element of {2,5} (non-simple family)
divisible by 2 if the length is odd, divisible by 5 if the length is even| |9|{3}{0}5|always divisible by some element of {2,5} (non-simple family)
divisible by 2 if the number of 3's is odd, divisible by 5 if the number of 3's is even| |11|2{5}|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |11|3{5}|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|3{7}|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|4{7}|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |11|{5}2|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |11|{5}3|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|{7}3|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|{7}4|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |14|4{0}1|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |14|B{0}1|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|3{D}|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|A{D}|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |14|1{0}B|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{D}3|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{4}9|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{8}5|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |8|6{4}7|always divisible by some element of {3,5,13}
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
(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}
divisible by 7 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 17 if the length is == 3 mod 4| |13|95{0}3|always divisible by some element of {5,7,17}
divisible by 7 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 17 if the length is == 0 mod 4| |16|{4}D|always divisible by some element of {3,7,13}
divisible by 3 if the length is == 0 mod 3, divisible by 7 if the length is == 2 mod 3, divisible by 13 if the length is == 1 mod 3| |16|{8}F|always divisible by some element of {3,7,13}
divisible by 3 if the length is == 1 mod 3, divisible by 7 if the length is == 0 mod 3, divisible by 13 if the length is == 2 mod 3| |17|7F{0}D|always divisible by some element of {3,5,29}
divisible by 3 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 29 if the length is == 3 mod 4| |17|D{0}7F|always divisible by some element of {3,5,29}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 29 if the length is == 0 mod 4| |20|8{0}1|always divisible by some element of {3,7}
divisible by 3 if the length is odd, divisible by 7 if the length is even| |20|D{0}1|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|7{J}|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|C{J}|always divisible by some element of {3,7}
divisible by 3 if the length is odd, divisible by 7 if the length is even| |20|1{0}D|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|{J}7|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |21|{7}D|always divisible by some element of {2,13,17}
divisible by 2 if the length is even, divisible by 13 if the length is == 1 mod 4, divisible by 17 if the length is == 3 mod 4| |23|7L{0}1|always divisible by some element of {3,5,53}
divisible by 3 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 53 if the length is == 3 mod 4| |23|1{0}7L|always divisible by some element of {3,5,53}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 53 if the length is == 0 mod 4| |27|JP{0}1|always divisible by some element of {5,7,73}
divisible by 7 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 73 if the length is == 3 mod 4| |27|1{0}JP|always divisible by some element of {5,7,73}
divisible by 7 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 73 if the length is == 0 mod 4| |30|A{0}9J|always divisible by some element of {7,13,19,31}
divisible by 7 if the length is == 0 mod 3, divisible by 13 if the length is == 1 mod 6, divisible by 19 if the length is == 2 mod 3, divisible by 31 if the length is == 0 mod 2| |32|A{0}1|always divisible by some element of {3,11}
divisible by 3 if the length is even, divisible by 11 if the length is odd| |32|N{0}1|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|9{V}|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|M{V}|always divisible by some element of {3,11}
divisible by 3 if the length is even, divisible by 11 if the length is odd| |32|1{0}N|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|{V}9|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|8{0}V|always divisible by some element of {3,5,41}
divisible by 3 if the length is odd, divisible by 5 if the length is == 0 mod 4, divisible by 41 if the length is == 2 mod 4| |34|6{0}1|always divisible by some element of {5,7}
divisible by 5 if the length is even, divisible by 7 if the length is odd| |34|5{X}|always divisible by some element of {5,7}
divisible by 5 if the length is odd, divisible by 7 if the length is even| |34|S{X}|always divisible by some element of {5,7}
divisible by 5 if the length is even, divisible by 7 if the length is odd| |34|{X}5|always divisible by some element of {5,7}
divisible by 5 if the length is odd, divisible by 7 if the length is even| |9|{1}|difference-of-squares factorization
(9^*n*−1)/8 = (3^*n*−1) × (3^*n*+1) / 8| |8|1{0}1|sum-of-cubes factorization
8^*n*+1 = (2^*n*+1) × (4^*n*−2^*n*+1)| |9|3{1}|difference-of-squares factorization
(25×9^*n*−1)/8 = (5×3^*n*−1) × (5×3^*n*+1) / 8| |9|3{8}|difference-of-squares factorization
4×9^*n*−1 = (2×3^*n*−1) × (2×3^*n*+1)| |9|{8}5|difference-of-squares factorization
9^*n*−4 = (3^*n*−2) × (3^*n*+2)| |9|3{8}35|difference-of-squares factorization
4×9^*n*−49 = (2×3^*n*−7) × (2×3^*n*+7)| |16|8{F}|difference-of-squares factorization
9×16^*n*−1 = (3×4^*n*−1) × (3×4^*n*+1)| |16|{F}7|difference-of-squares factorization
16^*n*−9 = (4^*n*−3) × (4^*n*+3)| |16|{4}1|difference-of-squares factorization
(4×16^*n*−49)/15 = (2×4^*n*−7) × (2×4^*n*+7) / 15| |16|B{4}1|difference-of-squares factorization
(169×16^*n*−49)/15 = (13×4^*n*−7) × (13×4^*n*+7) / 15| |16|1{5}|difference-of-squares factorization
(4×16^*n*−1)/3 = (2×4^*n*−1) × (2×4^*n*+1) / 3| |16|8{5}|difference-of-squares factorization
(25×16^*n*−1)/3 = (5×4^*n*−1) × (5×4^*n*+1) / 3| |16|10{5}|difference-of-squares factorization
(49×16^*n*−1)/3 = (7×4^*n*−1) × (7×4^*n*+1) / 3| |16|A1{5}|difference-of-squares factorization
(484×16^*n*−1)/3 = (22×4^*n*−1) × (22×4^*n*+1) / 3| |16|7{3}|difference-of-squares factorization
(36×16^*n*−1)/5 = (6×4^*n*−1) × (6×4^*n*+1) / 5| |16|3{F}AF|difference-of-squares factorization
4×16^*n*−81 = (2×4^*n*−9) × (2×4^*n*+9)| |16|30{F}AF|difference-of-squares factorization
49×16^*n*−81 = (7×4^*n*−9) × (7×4^*n*+9)| |16|3{F}A0F|difference-of-squares factorization
4×16^*n*−1521 = (2×4^*n*−39) × (2×4^*n*+39)| |16|30{F}A0F|difference-of-squares factorization
49×16^*n*−1521 = (7×4^*n*−39) × (7×4^*n*+39)| |16|{5}45|difference-of-squares factorization
(16^*n*−49)/3 = (4^*n*−7) × (4^*n*+7) / 3| |16|{C}B|difference-of-squares factorization
(4×16^*n*−9)/5 = (2×4^*n*−3) × (2×4^*n*+3) / 5| |16|{C}D|Aurifeuillian factorization of *x*⁴+4×*y*⁴
(4×16^*n*+1)/5 = (2×4^*n*−2×2^*n*+1) × (2×4^*n*+2×2^*n*+1) / 5| |16|{C}DD|Aurifeuillian factorization of *x*⁴+4×*y*⁴
(4×16^*n*+81)/5 = (2×4^*n*−6×2^*n*+9) × (2×4^*n*+6×2^*n*+9) / 5| |25|{1}|difference-of-squares factorization
(25^*n*−1)/24 = (5^*n*−1) × (5^*n*+1) / 24| |25|2{1}|difference-of-squares factorization
(49×25^*n*−1)/24 = (7×5^*n*−1) × (7×5^*n*+1) / 24| |25|5{1}|difference-of-squares factorization
(121×25^*n*−1)/24 = (11×5^*n*−1) × (11×5^*n*+1) / 24| |25|7{1}|difference-of-squares factorization
(169×25^*n*−1)/24 = (13×5^*n*−1) × (13×5^*n*+1) / 24| |25|C{1}|difference-of-squares factorization
(289×25^*n*−1)/24 = (17×5^*n*−1) × (17×5^*n*+1) / 24| |25|F{1}|difference-of-squares factorization
(361×25^*n*−1)/24 = (19×5^*n*−1) × (19×5^*n*+1) / 24| |25|M{1}|difference-of-squares factorization
(529×25^*n*−1)/24 = (23×5^*n*−1) × (23×5^*n*+1) / 24| |25|1F{1}|difference-of-squares factorization
(961×25^*n*−1)/24 = (31×5^*n*−1) × (31×5^*n*+1) / 24| |25|1{3}|difference-of-squares factorization
(9×25^*n*−1)/8 = (3×5^*n*−1) × (3×5^*n*+1) / 8| |25|1{8}|difference-of-squares factorization
(4×25^*n*−1)/3 = (2×5^*n*−1) × (2×5^*n*+1) / 3| |25|5{8}|difference-of-squares factorization
(16×25^*n*−1)/3 = (4×5^*n*−1) × (4×5^*n*+1) / 3| |25|A{3}|difference-of-squares factorization
(81×25^*n*−1)/8 = (9×5^*n*−1) × (9×5^*n*+1) / 8| |25|L{8}|difference-of-squares factorization
(64×25^*n*−1)/3 = (8×5^*n*−1) × (8×5^*n*+1) / 3| |25|{3}2|difference-of-squares factorization
(25^*n*−9)/8 = (5^*n*−3) × (5^*n*+3) / 8| |25|{8}3|difference-of-squares factorization
(25^*n*−16)/3 = (5^*n*−4) × (5^*n*+4) / 3| |25|{8}7|difference-of-squares factorization
(25^*n*−4)/3 = (5^*n*−2) × (5^*n*+2) / 3| |27|8{0}1|sum-of-cubes factorization
8×27^*n*+1 = (2×3^*n*+1) × (4×9^*n*−2×3^*n*+1)| |27|1{0}8|sum-of-cubes factorization
27^*n*+8 = (3^*n*+2) × (9^*n*−2×3^*n*+4)| |27|{D}E|sum-of-cubes factorization
(27^*n*+1)/2 = (3^*n*+1) × (9^*n*−3^*n*+1) / 2| |27|7{Q}|difference-of-cubes factorization
8×27^*n*−1 = (2×3^*n*−1) × (4×9^*n*+2×3^*n*+1)| |27|{Q}J|difference-of-cubes factorization
27^*n*−8 = (3^*n*−2) × (9^*n*+2×3^*n*+4)| |27|9{G}|difference-of-cubes factorization
(125×27^*n*−8)/13 = (5×3^*n*−2) × (25×9^*n*+10×3^*n*+4) / 13| |32|1{0}1|sum-of-5th-powers factorization
32^*n*+1 = (2^*n*+1) × (16^*n*−8^*n*+4^*n*−2^*n*+1)| |32|{1}|difference-of-5th-powers factorization
(32^*n*−1)/31 = (2^*n*−1) × (16^*n*+8^*n*+4^*n*+2^*n*+1) / 31| |36|3{7}|difference-of-squares factorization
(16×36^*n*−1)/5 = (4×6^*n*−1) × (4×6^*n*+1) / 5| |36|3{Z}|difference-of-squares factorization
4×36^*n*−1 = (2×6^*n*−1) × (2×6^*n*+1)| |36|8{Z}|difference-of-squares factorization
9×36^*n*−1 = (3×6^*n*−1) × (3×6^*n*+1)| |36|O{Z}|difference-of-squares factorization
25×36^*n*−1 = (5×6^*n*−1) × (5×6^*n*+1)| |36|{Z}B|difference-of-squares factorization
36^*n*−25 = (6^*n*−5) × (6^*n*+5)| |36|8{Z}B|difference-of-squares factorization
9×36^*n*−25 = (3×6^*n*−5) × (3×6^*n*+5)| |36|F{Z}B|difference-of-squares factorization
16×36^*n*−25 = (4×6^*n*−5) × (4×6^*n*+5)| |36|O{5}|difference-of-squares factorization
(169×36^*n*−1)/7 = (13×6^*n*−1) × (13×6^*n*+1) / 7| |36|O{7}|difference-of-squares factorization
(121×36^*n*−1)/5 = (11×6^*n*−1) × (11×6^*n*+1) / 5| |36|{9}1|difference-of-squares factorization
(9×36^*n*−289)/35 = (3×6^*n*−17) × (3×6^*n*+17) / 35| |36|T{9}1|difference-of-squares factorization
(1024×36^*n*−289)/35 = (32×6^*n*−17) × (32×6^*n*+17) / 35| |36|{K}H|difference-of-squares factorization
(4×36^*n*−25)/7 = (2×6^*n*−5) × (2×6^*n*+5) / 7| |36|{S}J|difference-of-squares factorization
(4×36^*n*−49)/5 = (2×6^*n*−7) × (2×6^*n*+7) / 5| |14|8{D}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×14^2×*n*−1 = (3×14^*n*−1) × (3×14^*n*+1)| |12|{B}9B|combine of factor 13 and difference-of-squares factorization
odd length is divisible by 13, even length has factorization 12^2×*n*−25 = (12^*n*−5) × (12^*n*+5)| |14|{D}5|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 14^2×*n*−9 = (14^*n*−3) × (14^*n*+3)| |17|1{9}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (25×17^2×*n*−9)/16 = (5×17^*n*−3) × (5×17^*n*+3) / 16| |17|7{9}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (121×17^2×*n*−9)/16 = (11×17^*n*−3) × (11×17^*n*+3) / 16| |17|{9}2|combine of factor 2 and difference-of-squares factorization
odd length is divisible by 2, even length has factorization (9×17^2×*n*−121)/16 = (3×17^*n*−11) × (3×17^*n*+11) / 16| |17|{9}8|combine of factor 2 and difference-of-squares factorization
odd length is divisible by 2, even length has factorization (9×17^2×*n*−25)/16 = (3×17^*n*−5) × (3×17^*n*+5) / 16| |19|1{6}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (4×19^2×*n*−1)/3 = (2×19^*n*−1) × (2×19^*n*+1) / 3| |19|{6}5|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization (19^2×*n*−4)/3 = (19^*n*−2) × (19^*n*+2) / 3| |19|7{2}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (64×19^2×*n*−1)/9 = (8×19^*n*−1) × (8×19^*n*+1) / 9| |19|89{6}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (484×19^2×*n*−1)/3 = (22×19^*n*−1) × (22×19^*n*+1) / 3| |24|3{N}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 4×24^2×*n*−1 = (2×24^*n*−1) × (2×24^*n*+1)| |24|5{N}|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 6×24^2×*n*+1−1 = (12×24^*n*−1) × (12×24^*n*+1)| |24|8{N}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×24^2×*n*−1 = (3×24^*n*−1) × (3×24^*n*+1)| |24|{6}1|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (6×24^2×*n*+1−121)/23 = (12×24^*n*−11) × (12×24^*n*+11) / 23| |24|{N}LN|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 24^2×*n*−49 = (24^*n*−7) × (24^*n*+7)| |33|F{W}|combine of factor 17 and difference-of-squares factorization
even length is divisible by 17, odd length has factorization 16×33^2×*n*−1 = (4×33^*n*−1) × (4×33^*n*+1)| |33|{W}H|combine of factor 17 and difference-of-squares factorization
odd length is divisible by 17, even length has factorization 33^2×*n*−16 = (33^*n*−4) × (33^*n*+4)| |33|3{P}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (121×33^2×*n*−25)/32 = (11×33^*n*−5) × (11×33^*n*+5) / 32| |33|D{P}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (441×33^2×*n*−25)/32 = (21×33^*n*−5) × (21×33^*n*+5) / 32| |34|1{B}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (4×34^2×*n*−1)/3 = (2×34^*n*−1) × (2×34^*n*+1) / 3| |34|8{X}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×34^2×*n*−1 = (3×34^*n*−1) × (3×34^*n*+1)| |34|{X}P|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 34^2×*n*−9 = (34^*n*−3) × (34^*n*+3)| 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
(4^*n*−1)/3 = (2^*n*−1) × (2^*n*+1) / 3| |8|{1}|difference-of-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(8^*n*−1)/7 = (2^*n*−1) × (4^*n*+2^*n*+1) / 7| |16|{1}|difference-of-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(16^*n*−1)/15 = (4^*n*−1) × (4^*n*+1) / 15| |27|{1}|difference-of-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(27^*n*−1)/26 = (3^*n*−1) × (9^*n*+3^*n*+1) / 26| |27|{G}7|difference-of-cubes factorization
but G7 is prime, and G7 is the only prime > *b* in this family
(8×27^*n*−125)/13 = (2×3^*n*−5) × (4×9^*n*+10×3^*n*+25) / 13| |36|{1}|difference-of-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(36^*n*−1)/35 = (6^*n*−1) × (6^*n*+1) / 35| 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 (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) 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 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.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://primes.utm.edu/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), 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*×*b*^*n*+*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, 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). This program uses the baby-step giant-step algorithm to find all primes *p* which divide *a*×*b*^*n*+*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*×*b*^*n*+*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 *x*⁴+4×*y*⁴), 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*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1 we only used it to sieve the sequence *a*×*b*^*n*+*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*^*n*+*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*^*n*+*c*, but 2 may not divide (*a*×*b*^*n*+*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, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64) or *PFGW* (https://sourceforge.net/projects/openpfgw/, https://primes.utm.edu/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*^*n*+*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, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64), 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, 28, 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*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the unproven probable prime| |:---|:---|:---| |11|1068|57₆₂₆₆₈|(57×11⁶²⁶⁶⁸−7)/10| |13|3194|C5₂₃₇₅₅C|(149×13²³⁷⁵⁶+79)/12| |13|3195|80₃₂₀₁₇111|8×13³²⁰²⁰+183| |13|3196|95₁₉₇₄₂₀|(113×13¹⁹⁷⁴²⁰−5)/12| |16|2345|DB₃₂₂₃₄|206×16³²²³⁴−11)/15| |16|2346|4₇₂₇₈₅DD|(4×16⁷²⁷⁸⁷+2291)/15| |16|2347|3₁₁₆₁₃₇AF|(16¹¹⁶¹³⁹+619)/5| |22|8003|BK₂₂₀₀₁5|(251×22²²⁰⁰²−335)/21| |28|25526|N6₂₄₀₅₁LR|(209×28²⁴⁰⁵³+3967)/9| |28|25527|5OA₃₁₂₃₈F|(4438×28³¹²³⁹+125)/27| |28|25528|O4O₉₄₅₃₅9|(6092×28⁹⁴⁵³⁶−143)/9| |30|2618|I0₂₄₆₀₈D|18×30²⁴⁶⁰⁹+13| 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*×*b*^*n*+*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|3165|50₂₇₀44|5×13²⁷²+56|http://factordb.com/cert.php?id=1100000002632397005| |13|3166|9₂₇₁095|(3×13²⁷⁴−6103)/4|http://factordb.com/cert.php?id=1100000003590431654| |13|3167|10₂₈₆7771|13²⁹⁰+16654|http://factordb.com/cert.php?id=1100000003590431633| |13|3168|9₃₀₈1|(3×13³⁰⁹−35)/4|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is 39/4×(13³⁰⁸−1), thus factor *N*−1 is equivalent to factor 13³⁰⁸−1, and for the factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=| |13|3169|B₃₄₁C4|(11×13³⁴³+61)/12|http://factordb.com/cert.php?id=1100000003590431618| |13|3170|8B₃₄₃|(107×13³⁴³−11)/12|http://factordb.com/cert.php?id=1100000002321018736| |13|3171|710₃₇₁111|92×13³⁷⁴+183|http://factordb.com/cert.php?id=1100000003590431609| |13|3172|75₃₇₅7|(89×13³⁷⁶+19)/12|http://factordb.com/cert.php?id=1100000003590431596| |13|3173|9B0₃₉₁9|128×13³⁹²+9|http://factordb.com/cert.php?id=1100000002632396790| |13|3174|7B0B₃₉₇|(15923×13³⁹⁷−11)/12|http://factordb.com/cert.php?id=1100000003590431574| |13|3175|10₄₁₄93|13⁴¹⁶+120|http://factordb.com/cert.php?id=1100000002523249240| |13|3176|81010₄₁₅1|17746×13⁴¹⁶+1|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is trivially fully factored| |13|3177|8110₄₃₅1|1366×13⁴³⁶+1|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is trivially fully factored| |13|3178|B7₄₈₆|(139×13⁴⁸⁶−7)/12|http://factordb.com/cert.php?id=1100000002321015892| |13|3179|B₅₆₃C|(11×13⁵⁶⁴+1)/12|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is 11/12×(13⁵⁶⁴−1), thus factor *N*−1 is equivalent to factor 13⁵⁶⁴−1, and for the factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=| |13|3180|1B₅₇₆|(23×13⁵⁷⁶−11)/12|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is 23/12×(13⁵⁷⁶−1), thus factor *N*−1 is equivalent to factor 13⁵⁷⁶−1, and for the factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=| |13|3181|80₆₉₃87|8×13⁶⁹⁵+111|http://factordb.com/cert.php?id=1100000002615636527| |13|3182|CC5₇₁₃|(2021×13⁷¹³−5)/12|http://factordb.com/cert.php?id=1100000002615627353| |13|3183|B₈₃₄74|(11×13⁸³⁶−719)/12|http://factordb.com/cert.php?id=1100000003590430871| |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|649|34D₇₀₈|47×14⁷⁰⁸−1|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html), since *N*+1 is trivially fully factored| |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|2329|D4₂₆₃D|(199×16²⁶⁴+131)/15|http://factordb.com/cert.php?id=1100000002468170238| |16|2330|E0₂₆₁4DD|14×16²⁶⁴+1245|http://factordb.com/cert.php?id=1100000003588388352| |16|2331|8C0₂₉₀ED|140×16²⁹²+237|http://factordb.com/cert.php?id=1100000003588388307| |16|2332|DA₃₀₅5|(41×16³⁰⁶−17)/3|http://factordb.com/cert.php?id=1100000003588388284| |16|2333|CE80₄₂₂D|3304×16⁴²³+13|http://factordb.com/cert.php?id=1100000003588388257| |16|2334|5F₅₄₄6F|6×16⁵⁴⁶−145|http://factordb.com/cert.php?id=1100000002604723967| |16|2335|88F₅₄₅|137×16⁵⁴⁵−1|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html), since *N*+1 is trivially fully factored| |16|2336|BE0₇₉₂BB|190×16⁷⁹⁴+187|http://factordb.com/cert.php?id=1100000003588387938| |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|3301|H₂₄₇A0H|(17×20²⁵⁰−59677)/19|http://factordb.com/cert.php?id=1100000003590502619| |20|3302|7₂₄₉A7|(7×20²⁵¹+1133)/19|http://factordb.com/cert.php?id=1100000003590502602| |20|3303|J7₂₇₀|(368×20²⁷⁰−7)/19|http://factordb.com/cert.php?id=1100000002325395462| |20|3304|J₃₃₀CCC7|20³³⁴−58953|http://factordb.com/cert.php?id=1100000003590502572| |20|3305|40₃₈₇404B|4×20³⁹¹+32091|http://factordb.com/cert.php?id=1100000003590502563| |20|3306|EC0₄₂₉7|292×20⁴³⁰+7|http://factordb.com/cert.php?id=1100000002633348702| |20|3307|G₄₄₇99|(16×20⁴⁴⁹−2809)/19|http://factordb.com/cert.php?id=1100000000840126753| |20|3308|3A₅₂₇3|(67×20⁵²⁸−143)/19|http://factordb.com/cert.php?id=1100000003590502531| |20|3309|E₅₆₆C7|(14×20⁵⁶⁸−907)/19|http://factordb.com/cert.php?id=1100000003590502516| |20|3310|JCJ₆₂₉|393×20⁶²⁹−1|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html), since *N*+1 is trivially fully factored| |20|3311|J₆₅₅05J|20⁶⁵⁸−7881|http://factordb.com/cert.php?id=1100000003590502490| |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|7984|I7G0₂₅₄H|8882×22²⁵⁵+17|http://factordb.com/cert.php?id=1100000003591372788| |22|7985|D0₂₅₅5EEF|13×22²⁵⁹+60339|http://factordb.com/cert.php?id=1100000003591371932| |22|7986|IK₃₂₂F|(398×22³²³−125)/21|http://factordb.com/cert.php?id=1100000000840384145| |22|7987|C0₃₄₀G9|12×22³⁴²+361|http://factordb.com/cert.php?id=1100000000840384159| |22|7988|77E₃₄₈K7|(485×22³⁵⁰+373)/3|http://factordb.com/cert.php?id=1100000003591369779| |22|7989|J₃₇₉KJ|(19×22³⁸¹+443)/21|http://factordb.com/cert.php?id=1100000003591369027| |22|7990|J₃₈₈EJ|(19×22³⁹⁰−2329)/21|http://factordb.com/cert.php?id=1100000003591367729| |22|7991|DJ₄₀₀|(292×22⁴⁰⁰−19)/21|http://factordb.com/cert.php?id=1100000002325880110| |22|7992|E₄₀₄K7|(2×22⁴⁰⁶+373)/3|http://factordb.com/cert.php?id=1100000003591366298| |22|7993|66F₄₅₃B3|(971×22⁴⁵⁵−705)/7|http://factordb.com/cert.php?id=1100000003591365809| |22|7994|L0₄₅₄B63|21×22⁴⁵⁷+5459|http://factordb.com/cert.php?id=1100000003591365331| |22|7995|L₄₈₃G3|22⁴⁸⁵−129|http://factordb.com/cert.php?id=1100000003591364730| |22|7996|E60₄₉₆L|314×22⁴⁹⁷+21|http://factordb.com/cert.php?id=1100000000632703239| |22|7997|I₆₂₆AF|(6×22⁶²⁸−1259)/7|http://factordb.com/cert.php?id=1100000000632724334| |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|3400|I0₂₄₁I5|18×24²⁴³+437|http://factordb.com/cert.php?id=1100000002633360037| |24|3401|D0₂₅₉KKD|13×24²⁶²+12013|http://factordb.com/cert.php?id=1100000003593270725| |24|3402|C7₂₉₈|(283×24²⁹⁸−7)/23|http://factordb.com/cert.php?id=1100000002326181235| |24|3403|20₃₁₃7|2×24³¹⁴+7|http://factordb.com/cert.php?id=1100000002355610241| |24|3404|BC0₃₃₁B|276×24³³²+11|http://factordb.com/cert.php?id=1100000002633359842| |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|25486|3₂₁₁M9|(28²¹³+4841)/9|http://factordb.com/cert.php?id=1100000003850161936| |28|25487|HD0₂₁₃D|489×28²¹⁴+13|http://factordb.com/cert.php?id=1100000003850161937| |28|25488|64O₂₁₇9|(1556×28²¹⁸−143)/9|http://factordb.com/cert.php?id=1100000000840840215| |28|25489|G0₂₁₇A0N|16×28²²⁰+7863|http://factordb.com/cert.php?id=1100000003850161938| |28|25490|55OA₂₂₆F|(110278×28²²⁷+125)/27|http://factordb.com/cert.php?id=1100000003850161939| |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|2613|AN₂₀₆|(313×30²⁰⁶−23)/29|http://factordb.com/cert.php?id=1100000002327651073| |30|2614|M₂₄₁QB|(22×30²⁴³+3139)/29|http://factordb.com/cert.php?id=1100000003593408295| |30|2615|M0₅₄₇SS7|22×30⁵⁵⁰+26047|http://factordb.com/cert.php?id=1100000003593407988| |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) (This data assumes that a number > 10²⁵⁰⁰⁰ which has passed the Miller–Rabin primality tests to all prime bases *p* < 64 and has passed the Baillie–PSW primality test and has trial factored to 2⁶⁴ 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) |*b*|number of minimal primes base *b*|base-*b* form of the top 10 known minimal prime base *b*|length of the top 10 known minimal prime base *b*|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the top 10 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
30301
14444
10103
3101
414|96
6
5
5
5
5
5
5
4
3|5⁹⁵+8
9391
3121
2341
2251
1951
1249
653
401
109|0|–| |6|11|40041
4441
4401
51
45
35
31
25
21
15|5
4
4
2
2
2
2
2
2
2|5209
1033
1009
31
29
23
19
17
13
11|0|–| |7|71|3₁₆1
510₇1
3₆01
1100021
531101
351101
300053
150001
100121
40054|17
10
8
7
6
6
6
6
6
5|(7¹⁷−5)/2
36×7⁸+1
(7⁸−47)/2
134471
91631
62819
50459
28813
16871
9643|0|–| |8|75|4₂₂₀7
5₁₃25
7₁₂1
77774₆1
74₇1
4₈1
55555025
5550525
5500525
4444477|221
15
13
11
9
9
8
7
7
7|(4×8²²¹+17)/7
(5×8¹⁵−173)/7
8¹³−7
(28669×8⁷−25)/7
(53×8⁸−25)/7
(4×8⁹−25)/7
11983381
1495381
1474901
1198399|0|–| |9|151|30₁₁₅₈11
27₆₈₆07
76₃₂₉2
561₃₆
10₂₅57
30₂₀51
8₁₉335
727₁₅07
51₁₃61
10₁₂507|1161
689
331
38
28
23
22
19
16
15|3×9¹¹⁶⁰+10
(23×9⁶⁸⁸−511)/8
(31×9³³⁰−19)/4
(409×9³⁶−1)/8
9²⁷+52
3×9²²+46
9²²−454
(527×9¹⁷−511)/8
(41×9¹⁵+359)/8
9¹⁴+412|0|–| |10|77|50₂₈27
5₁₁1
80555551
66600049
66000049
60000049
22000001
5200007
946669
666649|31
12
8
8
8
8
8
7
6
6|5×10³⁰+27
(5×10¹²−41)/9
80555551
66600049
66000049
60000049
22000001
5200007
946669
666649|0|–| |11|1068|57₆₂₆₆₈
557₁₀₁₁
7₇₅₉44
A₇₁₃58
85₂₂₀05
507₂₀₆
5₁₆₁2A
50₁₂₆57
10₁₂₅51
326₁₂₂|62669
1013
761
715
223
208
163
129
128
124|(57×11⁶²⁶⁶⁸−7)/10
(607×11¹⁰¹¹−7)/10
(7×11⁷⁶¹−367)/10
11⁷¹⁵−58
(17×11²²²−111)/2
(557×11²⁰⁶−7)/10
(11¹⁶³−57)/2
5×11¹²⁸+62
11¹²⁷+56
(178×11¹²²−3)/5|0|–| |12|106|40₃₉77
B0₂₇9B
B₆99B
AA000001
B00099B
AAA0001
BBBAA1
A00065
44AAA1
BBBB1|42
30
9
8
7
7
6
6
6
5|4×12⁴¹+91
11×12²⁹+119
12⁹−313
388177921
32847239
32555521
2985817
2488397
1097113
248821|0|–| |13|3196\~3197|95₁₉₇₄₂₀
80₃₂₀₁₇111
C5₂₃₇₅₅C
C₁₀₆₃₁92
B0₆₅₄₀BBA
390₆₂₆₆1
1770₂₇₀₃17
720₂₂₉₇2
930₁₅₅₁1
7₁₅₀₄1|197421
32021
23757
10633
6544
6269
2708
2300
1554
1505|(113×13¹⁹⁷⁴²⁰−5)/12
8×13³²⁰²⁰+183
(149×13²³⁷⁵⁶+79)/12
13¹⁰⁶³³−50
11×13⁶⁵⁴³+2012
48×13⁶²⁶⁷+1
267×13²⁷⁰⁵+20
93×13²²⁹⁸+2
120×13¹⁵⁵²+1
(7×13¹⁵⁰⁵−79)/12|1|254000| |14|650|4D₁₉₆₉₈
34D₇₀₈
8D₁₄₁85
8₈₆B
40₈₃49
8C₇₉3
18₇₉B
6B₇₇2B
4₆₃09
A₅₉3|19699
710
144
87
86
81
81
80
65
60|5×14¹⁹⁶⁹⁸−1
47×14⁷⁰⁸−1
9×14¹⁴³−79
(8×14⁸⁷+31)/13
4×14⁸⁵+65
(116×14⁸⁰−129)/13
(21×14⁸⁰+31)/13
(89×14⁷⁹−1649)/13
(4×14⁶⁵−667)/13
(10×14⁶⁰−101)/13|0|–| |15|1284|7₁₅₅97
E₁₄₅397
96₁₀₄08
7₇₃CE
7₅₉CCE
50₃₃17
EB₃₁
6330₂₆1
7050₂₄B
B70₂₄1|157
148
107
75
62
36
32
30
28
27|(15¹⁵⁷+59)/2
15¹⁴⁸−2558
(66×15¹⁰⁶−619)/7
(15⁷⁵+163)/2
(15⁶²+2413)/2
5×15³⁵+22
(207×15³¹−11)/14
1398×15²⁷+1
1580×15²⁵+11
172×15²⁵+1|0|–| |16|2347|3₁₁₆₁₃₇AF
4₇₂₇₈₅DD
DB₃₂₂₃₄
D0B₁₇₈₀₄
5BC₃₇₀₀D
90₃₅₄₂91
300F₁₉₆₀AF
20₁₇₁₃321
F8₁₅₁₇F
FAF₁₀₆₂45|116139
72787
32235
17806
3703
3545
1965
1717
1519
1066|(16¹¹⁶¹³⁹+619)/5
(4×16⁷²⁷⁸⁷+2291)/15
(206×16³²²³⁴−11)/15
(3131×16¹⁷⁸⁰⁴−11)/15
(459×16³⁷⁰¹+1)/5
9×16³⁵⁴⁴+145
769×16¹⁹⁶²−81
2×16¹⁷¹⁶+801
(233×16¹⁵¹⁸+97)/15
251×16¹⁰⁶⁴−187|0|–| |17|10408\~10428|570₅₁₃₁₀1
E9B₄₄₇₃₂
D0GD₃₇₀₉₆
G7₃₂₀₇₂F
150₂₄₃₂₅D
347₁₆₀₇₄
B30₁₃₀₇₇D
9D0₁₀₃₉₈5
10₉₀₁₉1F
B₉₀₁₅FB|51313
44734
37099
32074
24328
16076
13080
10401
9022
9017|92×17⁵¹³¹¹+1
(3963×17⁴⁴⁷³²−11)/16
(60381×17³⁷⁰⁹⁶−13)/16
(263×17³²⁰⁷³+121)/16
22×17²⁴³²⁶+13
(887×17¹⁶⁰⁷⁴−7)/16
190×17¹³⁰⁷⁸+13
166×17¹⁰³⁹⁹+5
17⁹⁰²¹+32
(11×17⁹⁰¹⁷+1077)/16|20|53000| |18|549|C0₆₂₆₈C5
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80₂₉₈B
C0₁₁₆F5
HD₉₃
GG0₃₀1
CF₃₀5
B₁₉6B
CCF₁₄5
7₁₄G7|6271
768
300
119
94
33
32
21
17
16|12×18⁶²⁷⁰+221
18⁷⁶⁸−37
8×18²⁹⁹+11
12×18¹¹⁸+275
(302×18⁹³−13)/17
304×18³¹+1
(219×18³¹−185)/17
(11×18²¹−1541)/17
(3891×18¹⁵−185)/17
(7×18¹⁶+2747)/17|0|–| |19|31410\~31435|4F0₄₉₈₄₇6
2₄₈₂₂₄7
2₄₅₈₈₆7A
90₄₂₉₉₄G
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10₂₂₇₉₀7717
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D₂₁₇₄₇88|49850
48225
45888
42996
36273
31091
26590
22795
22669
21749|91×19⁴⁹⁸⁴⁸+6
(19⁴⁸²²⁵+44)/9
(19⁴⁵⁸⁸⁸+926)/9
9×19⁴²⁹⁹⁵+16
(245×19³⁶²⁷²−11)/18
(20579×19³¹⁰⁸⁸−5)/18
(11×19²⁶⁵⁹⁰+1447)/18
19²²⁷⁹⁴+50566
(223×19²²⁶⁶⁸+83)/18
(13×19²¹⁷⁴⁹−1813)/18|25|50000| |20|3314|G0₆₂₆₉D
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2450
1166
658
631
568
529
449
432
392|16×20⁶²⁷⁰+13
(241×20²⁴⁴⁹−13)/19
5×20¹¹⁶⁵+219
20⁶⁵⁸−7881
393×20⁶²⁹−1
(14×20⁵⁶⁸−907)/19
(67×20⁵²⁸−143)/19
(16×20⁴⁴⁹−2809)/19
292×20⁴³⁰+7
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17141
11398
11108
10091
8261
5353
3796
2335
2162|118×21¹⁹⁸⁴⁹+1
245×21¹⁷¹³⁹+74
21¹¹³⁹⁷+118
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322×21⁸²⁵⁹+11
(53×21⁵³⁵²−233)/20
(369×21³⁷⁹⁵+71)/20
8586×21²³³²+17
19×21²¹⁶¹+165982|22|20000| |22|8003|BK₂₂₀₀₁5
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3817
2388
961
772
764
628
499
485
458|(251×22²²⁰⁰²−335)/21
(22³⁸¹⁷−289)/3
22²³⁸⁸−653
(22⁹⁶¹+857)/3
19×22⁷⁷¹+199779
20×22⁷⁶³+7041
(6×22⁶²⁸−1259)/7
314×22⁴⁹⁷+21
22⁴⁸⁵−129
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18402
16363
15740
14051
13801
13156
12459
12412
11999|(7×23¹⁹⁰⁶⁹−2119)/22
3716×23¹⁸³⁹⁹+9
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458×23¹⁵⁷³⁸+17
122×23¹⁴⁰⁴⁹+1
(21×23¹³⁸⁰¹−241)/22
10483×23¹³¹⁵³+14
9444×23¹²⁴⁵⁶+19
(19×23¹²⁴¹¹−507)/2
(21×23¹¹⁹⁹⁹−8×23⁷−13)/22|132|20000 |24|3409|N00N₈₁₂₉LN
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5953
2955
2700
2647
334
315
299
263
244|13249×24⁸¹³¹−49
201×24⁵⁹⁵¹−1
10×24²⁹⁵⁴+5053
(13×24²⁷⁰⁰+4403)/23
24²⁶⁴⁷−1201
276×24³³²+11
2×24³¹⁴+7
(283×24²⁹⁸−7)/23
13×24²⁶²+12013
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19802
19476
18510
18470
18247
18162
18152
17964
17959|292×25¹⁹⁸⁶⁸+3711
(3569×25¹⁹⁸⁰⁰+163)/12
207538×25¹⁹⁴⁷²+19
(57317×25¹⁸⁵⁰⁷−21)/8
(1601×25¹⁸⁴⁶⁸+923)/4
(7×25¹⁸²⁴⁷−711)/8
25¹⁸¹⁶¹+207544
32×25¹⁸¹⁵⁰+8469
7390×25¹⁷⁹⁶¹+7
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44307
23302
20978
20281
19393
15924
8773
4367
4223|22×26⁶¹¹⁸⁹+1649
19×26⁴⁴³⁰⁶+13843
(34×26²³³⁰¹−79)/5
559×26²⁰⁹⁷⁶+7
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(26¹⁹³⁹³+179)/5
(32569×26¹⁵⁹²¹+21)/5
(22×26⁸⁷⁷³+53)/25
20×26⁴³⁶⁶+473
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17471
17277
16476
16231
15938
15197
14710
13560
13201|(16×27¹⁹³⁹⁶+23)/13
(541×27¹⁷⁴⁷⁰−151)/26
26×27¹⁷²⁷⁶+6727
(191×27¹⁶⁴⁷⁵+173)/26
(493×27¹⁶²³⁰−18043)/26
27¹⁵⁹³⁷+482
(563×27¹⁵¹⁹⁶−277)/26
(301×27¹⁴⁷⁰⁹+167)/26
(144629×27¹³⁵⁵⁷−17)/26
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24054
5271
4242
3748
1902
1425
1372
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16×28¹⁹⁰¹+303
(10×28¹⁴²⁵−2899)/27
(17×28¹³⁷¹−11)/3
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24610
4883
1024
551
243
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155
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22×30⁵⁵⁰+26047
(22×30²⁴³+3139)/29
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(19×30¹⁵⁵+6071)/29
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19374
18514
18218
18064
17029
17010
16804
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10119
7261
6180
5780
4621
4564
3935
3926
3427|(53×36²⁶⁵⁶⁸+101)/7
(19×36¹⁰¹¹⁹+2501)/35
1137×36⁷²⁵⁹+19
528×36⁶¹⁷⁸+31
16×36⁵⁷⁷⁹−1163
(36549×36⁴⁶¹⁹−289)/35
(979×36⁴⁵⁶³−629)/35
25×36³⁹³⁴−901
(4×36³⁹²⁶+941)/35
(81904×36³⁴²⁵−49)/5|6|50000| 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*×*b*^*n*±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