Name already in use

Primality of these 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, http://factordb.com/listtype.php?t=1) in the minimal sets for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases b with ≤ 10 unsolved families) are not certificated yet (technically, probable primality tests (https://en.wikipedia.org/wiki/Probabilistic_algorithm) such as 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, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm) and Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html) are used for these numbers (and thus these numbers are only probable primes and not definitely primes (https://en.wikipedia.org/wiki/Provable_prime, http://factordb.com/listtype.php?t=4), i.e. they might be pseudoprimes (https://en.wikipedia.org/wiki/Pseudoprime, https://primes.utm.edu/glossary/xpage/Pseudoprime.html, https://www.rieselprime.de/ziki/Pseudoprime, https://mathworld.wolfram.com/Pseudoprime.html)), because of their sizes and neither N−1 nor N+1 can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html) (i.e. they are ordinary primes (https://primes.utm.edu/glossary/xpage/OrdinaryPrime.html)), 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 (the CPU time (https://en.wikipedia.org/wiki/CPU_time) is longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe), even if we can do 10⁹ bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second)) to run on these numbers, the only known primality test with polynomial time (https://en.wikipedia.org/wiki/Time_complexity#Polynomial_time, https://mathworld.wolfram.com/PolynomialTime.html) of the number of digits is AKS primality test (https://en.wikipedia.org/wiki/AKS_primality_test, https://mathworld.wolfram.com/AKSPrimalityTest.html, https://primes.utm.edu/prove/prove4_3.html, http://cr.yp.to/papers/aks-20030125-retypeset20220327.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_70.pdf)), but it has time complexity (https://en.wikipedia.org/wiki/Time_complexity) O(log(n)¹²) (where O is the big O notation (https://en.wikipedia.org/wiki/Big_O_notation, https://primes.utm.edu/glossary/xpage/BigOh.html, https://mathworld.wolfram.com/Big-ONotation.html), log is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://primes.utm.edu/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html)) and too large, also a near-polynomial time (although not completely polynomial time) primality test, APR primality test (https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://www.rieselprime.de/ziki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://mathworld.wolfram.com/Adleman-Pomerance-RumelyPrimalityTest.html, https://primes.utm.edu/prove/prove4_1.html), which has time complexity O(log(n)^{log(log(log(n)))}) and also too large), all of these probable primes are > 10²⁵⁰⁰⁰, if they are in fact primes, then they are minimal primes to the corresponding bases.

If we want to use the classical tests (https://primes.utm.edu/prove/prove3.html) to prove the primality of a large probable prime N, then we must factor N−1 or N+1 to the factored part ≥ 1/4, and except 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) with the primes up to certain limit (say 10¹⁶) and the algebra factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares), sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html), Aurifeuillian factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html) of x⁴+4×y⁴, and the algebra factors of the Cunningham number (https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, https://en.wikipedia.org/wiki/The_Cunningham_project, https://primes.utm.edu/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, http://myfactors.mooo.com/, https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)) bⁿ±1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm and https://brnikat.com/nums/cullen_woodall/algebraic.txt)), we can use the elliptic-curve factorization method (https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization, https://www.rieselprime.de/ziki/Elliptic_curve_method, https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html, http://factordb.com/listecm.php?c=1, http://www.loria.fr/~zimmerma/ecmnet/, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html) or the Pollard P−1 method (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, http://factordb.com/listecm.php?c=2) or the Williams P+1 method (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, http://factordb.com/listecm.php?c=3) or the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html), to factor these numbers (see http://www.numericana.com/answer/factoring.htm), elliptic-curve factorization method and Pollard P−1 method and Williams P+1 method are methods which find a non-large (say between 10²⁰ and 10⁹⁰) prime factor, an elliptic-curve factorization program is GMP-ECM (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM), special number field sieve and general number field sieve are methods which factor a large number (say between 10¹⁰⁰ and 10⁴⁰⁰), special number field sieve can only be used on a number of special form, e.g. numbers of the form a×bⁿ+c with small a, b, c, the difficulty (https://www.rieselprime.de/ziki/SNFS_polynomial_selection) of such a number is equivalent to general number field sieve for a general number around b^3×n/2, a general number field sieve program is GGNFS (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS), also a (special or general) number field program is CADO-NFS (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0), however, all these factorization algorithms (https://en.wikipedia.org/wiki/Algorithm) take long time, i.e. they cannot be done in polynomial time (https://en.wikipedia.org/wiki/Time_complexity#Polynomial_time, https://mathworld.wolfram.com/PolynomialTime.html) (unlike primality proving, when the numbers are sufficiently large, no efficient, non-quantum (https://en.wikipedia.org/wiki/Quantum_computer) integer factorization algorithm is known, i.e. integer factorization may be P-complete (https://en.wikipedia.org/wiki/P-complete) and NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) and NP-hard (https://en.wikipedia.org/wiki/NP-hard, https://mathworld.wolfram.com/NP-HardProblem.html) (thus, factor a large integer is much harder than determining whether an integer of the same size is prime (determining whether an integer is prime and factor an integer are two completely different problems, we can quickly use Fermat primality test (https://en.wikipedia.org/wiki/Fermat_primality_test, https://mathworld.wolfram.com/FermatCompositenessTest.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://en.wikipedia.org/wiki/Fermat%27s_little_theorem, https://primes.utm.edu/glossary/xpage/FermatsLittleTheorem.html, https://mathworld.wolfram.com/FermatsLittleTheorem.html, https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html, https://primes.utm.edu/prove/prove2_2.html) to prove that an integer is composite, although the most ancient 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) and sieve of Eratosthenes (https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes, https://primes.utm.edu/glossary/xpage/SieveOfEratosthenes.html, https://www.rieselprime.de/ziki/Sieve_of_Eratosthenes, https://mathworld.wolfram.com/SieveofEratosthenes.html) can solving these two problems simultaneously), there are many numbers with 500 digits to 10000 digits which are known to be composite but do not have any known factors other than 1 and themselves). However, it has not been proven that no efficient algorithm exists (this is indeed an unsolved problem in computer science (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science)). The presumed difficulty (https://en.wikipedia.org/wiki/Computational_hardness_assumption) of this problem is at the heart of widely used algorithms in cryptography (https://en.wikipedia.org/wiki/Cryptography, https://mathworld.wolfram.com/Cryptography.html) such as RSA (https://en.wikipedia.org/wiki/RSA_(cryptosystem), https://primes.utm.edu/glossary/xpage/RSA.html, https://mathworld.wolfram.com/RSAEncryption.html, https://web.archive.org/web/20061209135708/http://www.rsasecurity.com/rsalabs/node.asp?id=2093), there are many large semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://primes.utm.edu/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html), called RSA numbers (https://en.wikipedia.org/wiki/RSA_numbers, https://primes.utm.edu/glossary/xpage/RSAExample.html, https://mathworld.wolfram.com/RSANumber.html), which are very hard to factor and are part of the RSA Factoring Challenge (https://en.wikipedia.org/wiki/RSA_Factoring_Challenge). Besides, integer factorization can be used for public-key cryptography (https://en.wikipedia.org/wiki/Public-key_cryptography, https://primes.utm.edu/glossary/xpage/PublicKey.html, https://mathworld.wolfram.com/Public-KeyCryptography.html) is because it has no known polynomial time algorithm. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves (https://en.wikipedia.org/wiki/Elliptic_curve, https://mathworld.wolfram.com/EllipticCurve.html), algebraic number theory (https://en.wikipedia.org/wiki/Algebraic_number_theory, https://mathworld.wolfram.com/AlgebraicNumberTheory.html), and quantum computing (https://en.wikipedia.org/wiki/Quantum_computing)), and hence to do this is impractically.

Thus, for these numbers, we usually use the general purpose tests (https://primes.utm.edu/prove/prove4.html), which do not require factorization of N±1 and can be used for ordinary primes (https://primes.utm.edu/glossary/xpage/OrdinaryPrime.html).

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 (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 10¹⁶ (thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html)), and since all of them are > 10²⁵⁰⁰⁰, thus the property that they are in fact composite is less than 10⁻²⁰⁰⁰ (but still not zero!), see https://primes.utm.edu/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf) (bases 11, 16, 22, 30 have no unsolved families but have unproven PRPs, thus, we can 99.999999...999999% (with >2000 9's) say that the minimal sets for bases 11, 16, 22, 30 are these four sets, but we still cannot 100% say this)

A reference of many types of pseudoprimes: http://ntheory.org/pseudoprimes.html; also a reference of data for all base 2 Fermat pseudoprimes < 2⁶⁴: http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html; also examples of strong pseudoprimes to many bases: https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1192971-8/S0025-5718-1993-1192971-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_41.pdf), https://arxiv.org/pdf/1207.0063.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_42.pdf), https://arxiv.org/pdf/1509.00864.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_43.pdf), https://www.sciencedirect.com/science/article/pii/S0747717185710425?via%3Dihub (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_44.pdf), https://primes.utm.edu/curios/page.php?number_id=4265, https://primes.utm.edu/prove/prove2_3.html, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773; also a research of pseudoprimes: http://www.numericana.com/answer/pseudo.htm

• If 57₆₂₆₆₈ (base 11) is in fact composite, then 5{7} will be an additional unsolved family for base 11.
• If C5₂₃₇₅₅C (base 13) is in fact composite, then C{5}C will be an additional unsolved family for base 13.
• If 80₃₂₀₁₇111 (base 13) is in fact composite, then 8{0}111 will be an additional unsolved family for base 13.
• If 95₁₉₇₄₂₀ (base 13) is in fact composite, then 9{5} will be an additional unsolved family for base 13.
• If DB₃₂₂₃₄ (base 16) is in fact composite, then D{B} will be an additional unsolved family for base 16.
• If 4₇₂₇₈₅DD (base 16) is in fact composite, then {4}DD will be an additional unsolved family for base 16.
• If 3₁₁₆₁₃₇AF (base 16) is in fact composite, then {3}AF will be an additional unsolved family for base 16.
• If BK₂₂₀₀₁5 (base 22) is in fact composite, then B{K}5 will be an additional unsolved family for base 22.
• If 5₁₉₃₉₁6F (base 26) is in fact composite, then {5}6F will be an additional unsolved family for base 26.
• If 7₂₀₂₇₉OL (base 26) is in fact composite, then {7}OL will be an additional unsolved family for base 26.
• If LD0₂₀₉₇₅7 (base 26) is in fact composite, then LD{0}7 will be an additional unsolved family for base 26.
• If 6K₂₃₃₀₀5 (base 26) is in fact composite, then 6{K}5 will be an additional unsolved family for base 26.
• If J0₄₄₃₀₃KCB (base 26) is in fact composite, then J{0}KCB will be an additional unsolved family for base 26.
• If M0₆₁₁₈₆2BB (base 26) is in fact composite, then M{0}2BB will be an additional unsolved family for base 26.
• If N6₂₄₀₅₁LR (base 28) is in fact composite, then N{6}LR will be an additional unsolved family for base 28.
• If 5OA₃₁₂₃₈F (base 28) is in fact composite, then 5O{A}F will be an additional unsolved family for base 28 (unless a prime in the family O{A}F is known, since the family 5O{A}F is covered by the unsolved family O{A}F (if the prime in the family 5O{A}F has longer length than the prime in the family O{A}F) (i.e. the family O{A}F is a subfamily of the family 5O{A}F), and there is still a possibility that the smallest prime in the family 5O{A}F has longer length than the smallest prime in the family O{A}F, thus we still cannot definitely say that there are 25529 minimal primes in base 28 even if we assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://primes.utm.edu/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html) that all unsolved families have a prime, this is reasonable, references: https://primes.utm.edu/mersenne/heuristic.html, https://primes.utm.edu/notes/faq/NextMersenne.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes).
• If O4O₉₄₅₃₅9 (base 28) is in fact composite, then O4{O}9 will be an additional unsolved family for base 28.
• If I0₂₄₆₀₈D (base 30) is in fact composite, then I{0}D will be an additional unsolved family for base 30.
• If 7K₂₆₅₆₇Z (base 36) is in fact composite, then 7{K}Z will be an additional unsolved family for base 36.
• If S0₇₅₀₀₇8H (base 36) is in fact composite, then S{0}8H will be an additional unsolved family for base 36.
• If P₈₁₉₉₃SZ (base 36) is in fact composite, then {P}SZ will be an additional unsolved family for base 36.

For the files in this page:

• File "unprovenPRP b": The unproven probable primes which are minimal primes in base b (assuming their primality), expressed as base b strings with x(y^n)z format (which in the README files it is denoted as "xy_nz") (y^n means n copies of y, e.g. 34(5^6)78 is 3455555578)
• File "primoinput b n": The input file for Primo for the nth minimal prime in base b (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, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) (local copy from factordb (http://factordb.com/)), they should be renamed to ".in" files before inputting to Primo, e.g. file "primoinput11_1068" is the input file for Primo for the 1068th minimal prime in base 11, i.e. the input file for Primo for the PRP 57₆₂₆₆₈ in base 11, which equals the PRP (57×11⁶²⁶⁶⁸−7)/10. (Note: There is no input file for Primo for the PRP 95₁₉₇₄₂₀ in base 13 (which equals the PRP (113×13¹⁹⁷⁴²⁰−5)/12) since this PRP is too large (>10¹⁴⁹⁹⁹⁹) to be PRP-tested in factordb, and factordb does not have input file for Primo for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has input file for Primo for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))