(cache)minimal-elements-of-the-prime-numbers/primality-certificates/README.md at main · xayahrainie4793/minimal-elements-of-the-prime-numbers

(Edit: Now you can click http://5.199.134.130/certificates.tar.xz and http://5.199.134.130/certificates.tar.xz.SHA256SUM and http://5.199.134.130/certificates.tar.xz.par2 and http://5.199.134.130/certificates.tar.xz.vol00+10.par2 to download all primality certificates in factordb (which includes the primality certificates of the minimal primes in bases 2 ≤ b ≤ 36 in factordb), also the .xz files in http://5.199.134.130/certificates/ ("n.xz" for the primality certificates of the definitely primes which are proven primes by primality certificates (i.e. not proven primes by N−1 or N+1 or combine of N−1 and N+1) with n decimal digits for n ≥ 300) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"))

(In progess to add bases b = 17 and b = 21)

The large minimal primes in base b are of the form (a×bⁿ+c)/gcd(a+c,b−1) for some a, b, c, n such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1 (i.e. they are the near-Cunningham numbers (http://factordb.com/tables.php?open=4, https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10")), the large numbers (i.e. the numbers with large n, say n > 1000) can be easily proven primes using N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) if and only if c = ±1 and gcd(a+c,b−1) = 1 (if this large minimal prime in base b is xy_nz (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) in base b, then c = 1 and gcd(a+c,b−1) = 1 if and only if the digit y is 0 and the string z is 1, and c = −1 and gcd(a+c,b−1) = 1 if and only if the digit y is b−1 and the string z is 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), if we reduce these families by removing all trailing digits y from x, and removing all leading digits y from z, to make the families be easier, e.g. family 12333{3}33345 in base b is reduced to family 12{3}45 in base b, since they are in fact the same family), except this special case (https://en.wikipedia.org/wiki/Special_case) of c = ±1 and gcd(a+c,b−1) = 1, such numbers need primality certificates to be proven primes (otherwise, they will only be probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1)), and elliptic curve primality proving are used for these numbers.

(for more examples of numbers which are proven primes using the N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or the N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), see https://www.mersenneforum.org/showthread.php?t=16209)

the 3176th minimal prime in base 13, 81010₄₁₅1, which equals 17746×13⁴¹⁶+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000003590431555, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000003590431556&open=ecm
the 3177th minimal prime in base 13, 8110₄₃₅1, which equals 1366×13⁴³⁶+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000002373259109, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000002373259124&open=ecm
the 3188th minimal prime in base 13, 930₁₅₅₁1, which equals 120×13¹⁵⁵²+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961452, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000765961453&open=ecm
the 3191st minimal prime in base 13, 390₆₂₆₆1, which equals 48×13⁶²⁶⁷+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961441, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000765961451&open=ecm
the 649th minimal prime in base 14, 34D₇₀₈, which equals 47×14⁷⁰⁸−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000001540144903, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000001540144907&open=ecm
the 650th minimal prime in base 14, 4D₁₉₆₉₈, which equals 5×14¹⁹⁶⁹⁸−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000884560233, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000000884560625&open=ecm
the 2335th minimal prime in base 16, 88F₅₄₅, which equals 137×16⁵⁴⁵−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000413679658, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000000413877337&open=ecm
the 10317th minimal prime in base 17, 5A70₂₇₄1, which equals 1622×17²⁷⁵+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000003782940709, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000003782941930&open=ecm
the 10359th minimal prime in base 17, 9D0₁₀₆₇1, which equals 166×17¹⁰⁶⁸+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961369, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000765961370&open=ecm
the 10370th minimal prime in base 17, A0₁₃₅₅1, which equals 10×17¹³⁵⁶+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000034167087, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000271866825&open=ecm
the 10386th minimal prime in base 17, 530₄₈₆₇1, which equals 88×17⁴⁸⁶⁸+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000762660735, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000762660737&open=ecm
the 10408th minimal prime in base 17, 570₅₁₃₁₀1, which equals 92×17⁵¹³¹¹+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961389, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000785469616&open=ecm
the 3310th minimal prime in base 20, JCJ₆₂₉, which equals 393×20⁶²⁹−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000001559454258, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000001559454271&open=ecm
the 13373rd minimal prime in base 21, 5D0₁₉₈₄₈1, which equals 118×21¹⁹⁸⁴⁹+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000777265872, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000785469310&open=ecm
the 3408th minimal prime in base 24, 88N₅₉₅₁, which equals 201×24⁵⁹⁵¹−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000003593275880, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000003593373246&open=ecm
the 25509th minimal prime in base 28, EB0₄₀₅1, which equals 403×28⁴⁰⁶+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000001534442374, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000001534442380&open=ecm
the 2616th minimal prime in base 30, C0₁₀₂₂1, which equals 12×30¹⁰²³+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000785448736, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000785448737&open=ecm
the 2619th minimal prime in base 30, OT₃₄₂₀₅, which equals 25×30³⁴²⁰⁵−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000800812865, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000000819405041&open=ecm
the 35237th minimal prime in base 36, P8Z₃₉₀, which equals 909×36³⁹⁰−1, N+1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000764100228, for the factorization of N+1 in factordb see http://factordb.com/index.php?id=1100000000764100231&open=ecm

(these primes can be proven prime using Yves Gallot's Proth.exe (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see the README file for LLR (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html))

Also, there are no primality certificates for these primes in factordb because although they are > 10²⁹⁹, but their N−1 or N+1 is fully factored (but not trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored, however, only need trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹²) and the largest prime factor is < 10²⁹⁹ (primes < 10²⁹⁹ are automatically proven primes in factordb): (i.e. their N−1 or N+1 are product of a 10¹²-smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) and a prime < 10²⁹⁹) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of N−1 or N+1 is < 10²⁹⁹, and the second-greatest prime factor (https://oeis.org/A087039, https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):")) of this number (N−1 or N+1) is < 10¹²)

the 2328th minimal prime in base 16, 880₂₄₆7, with 300 decimal digits, N−1 is 2³ × 3 × 7 × 13 × 25703261 × (289-digit prime)
the 10311st minimal prime in base 17, 85A₂₄₁55, with 302 decimal digits, N+1 is 2 × 1291 × 942385161439 × (286-digit prime)
the 10312nd minimal prime in base 17, 90₂₄₂701, with 303 decimal digits, N−1 is 2⁶ × 17² × 1773259 × 4348181 × 603217519 × (277-digit prime)
the 10315th minimal prime in base 17, E7₂₅₅A, with 317 decimal digits, N−1 is 2⁴ × 283 × 619471 × 62754967151 × (296-digit prime)
the 25174th minimal prime in base 26, OL0₂₁₄M9, with 309 decimal digits, N−1 is 2² × 5² × 7 × 223 × 42849349 × (296-digit prime)
the 25485th minimal prime in base 28, JN₂₀₆, with 300 decimal digits, N−1 is 2 × 1061 × 1171 × 74311 × (289-digit prime)

The helper file for the 2328th minimal prime in base 16 (880₂₄₆7) in factordb: http://factordb.com/helper.php?id=1100000002468140199

The helper file for the 10311st minimal prime in base 17 (85A₂₄₁55) in factordb: http://factordb.com/helper.php?id=1100000003782940703

The helper file for the 10312nd minimal prime in base 17 (90₂₄₂701) in factordb: http://factordb.com/helper.php?id=1100000003782940704

The helper file for the 10315th minimal prime in base 17 (E7₂₅₅A) in factordb: http://factordb.com/helper.php?id=1100000003782940707

The helper file for the 25174th minimal prime in base 26 (OL0₂₁₄M9) in factordb: http://factordb.com/helper.php?id=1100000000840631576

The helper file for the 25485th minimal prime in base 28 (JN₂₀₆) in factordb: http://factordb.com/helper.php?id=1100000002611724435

Factorization of N−1 for the 2328th minimal prime in base 16 (880₂₄₆7) in factordb: http://factordb.com/index.php?id=1100000002468140641&open=ecm

Factorization of N+1 for the 10311st minimal prime in base 17 (85A₂₄₁55) in factordb: http://factordb.com/index.php?id=1100000003782944423&open=ecm

Factorization of N−1 for the 10312nd minimal prime in base 17 (90₂₄₂701) in factordb: http://factordb.com/index.php?id=1100000003782941925&open=ecm

Factorization of N−1 for the 10315th minimal prime in base 17 (E7₂₅₅A) in factordb: http://factordb.com/index.php?id=1100000003782941928&open=ecm

Factorization of N−1 for the 25174th minimal prime in base 26 (OL0₂₁₄M9) in factordb: http://factordb.com/index.php?id=1100000000840631577&open=ecm

Factorization of N−1 for the 25485th minimal prime in base 28 (JN₂₀₆) in factordb: http://factordb.com/index.php?id=1100000002611724440&open=ecm

(for the examples of generalized repunit primes (all generalized repunit primes base b have that N−1 is product of a Cunningham number (base b, the −1 side) and a small number (namely b/(b−1))), see https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html and https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html and https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html and http://xenon.stanford.edu/~tjw/pp/index.html)

(for more examples see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "proven@" in the "note" column), also see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 for the related numbers (although not all of them are related to Cunningham numbers), e.g. "11101_4809" (decimal (base b = 10) form: 1₄₈₀₇01, algebraic form: (10⁴⁸⁰⁹−91)/9) is related to "Phi_4807_10" (the number Φ₄₈₀₇(10), where Φ is the cyclotomic polynomial), "15551_2197" (decimal (base b = 10) form: 15₂₁₉₆1, algebraic form: (14×10²¹⁹⁷−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base b = 10) form: 93₂₁₉₅07, algebraic form: (28×10²¹⁹⁷−79)/3), "16667_4296" (decimal (base b = 10) form: 16₄₂₉₅7, algebraic form: (5×10⁴²⁹⁶+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base b = 10) form: 3₁₂₈₉₀7, algebraic form: (10¹²⁸⁹¹+11)/3), "20111_2692" (decimal (base b = 10) form: 201₂₆₉₂, algebraic form: (181×10²⁶⁹²−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base b = 10) form: 201₂₆₉₃, algebraic form: (181×10²⁶⁹³−1)/9), "23309_10029" (decimal (base b = 10) form: 23₁₀₀₂₇09, algebraic form: (7×10¹⁰⁰²⁹−73)/3) is related to "Phi_5014_10" (the number Φ₅₀₁₄(10), where Φ is the cyclotomic polynomial), "37773_15768" (decimal (base b = 10) form: 37₁₅₇₆₇3, algebraic form: (34×10¹⁵⁷⁶⁸−43)/9) is related to "Phi_7884_10" (the number Φ₇₈₈₄(10), where Φ is the cyclotomic polynomial), "6805w7_3739" (decimal (base b = 10) form: 6805₃₇₃₈7, algebraic form: (6125×10³⁷³⁹+13)/9, the prime is a cofactor of it (divided it by 3²)) is related to "27227_3741" (decimal (base b = 10) form: 272₃₇₄₀7, algebraic form: (245×10³⁷⁴¹+43)/9), "68883_5132" (decimal (base b = 10) form: 68₅₁₃₁3, algebraic form: (62×10⁵¹³²−53)/9) is related to "Phi_1283_10" (the number Φ₁₂₈₃(10), where Φ is the cyclotomic polynomial), "79921_11629" (decimal (base b = 10) form: 79₁₁₆₂₇21, algebraic form: 8×10¹¹⁶²⁹−79) is related to "Phi_2907_10" (the number Φ₂₉₀₇(10), where Φ is the cyclotomic polynomial), "80081_5736" (decimal (base b = 10) form: 80₅₇₃₄81, algebraic form: 8×10⁵⁷³⁶+81) is related to "Phi_11470_10" (the number Φ₁₁₄₇₀(10), where Φ is the cyclotomic polynomial), "83w16w7_543" (decimal (base b = 10) form: 83₅₄₂16₅₄₂7, algebraic form: (25×10¹⁰⁸⁶−5×10⁵⁴³+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base b = 10) form: 1₃₂₅₆03, algebraic form: (10³²⁵⁸−73)/9), etc. the N−1 of "11101_4809" is 100 × R₄₈₀₇(10) (which is equivalent to the Cunningham number 10⁴⁸⁰⁷−1) and Φ₄₈₀₇(10) is an algebraic factor of the Cunningham number 10⁴⁸⁰⁷−1, the N−1 of "93307_2197" is 6 × "15551_2197", the N−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the N−1 of "20111_2693" is 10 × "20111_2692", the N+1 of "23309_10029" is 210 × R₁₀₀₂₈(10) (which is equivalent to the Cunningham number 10¹⁰⁰²⁸−1) and Φ₅₀₁₄(10) is an algebraic factor of the Cunningham number 10¹⁰⁰²⁸−1, the N+1 of "37773_15768" is 34 × R₁₅₇₆₈(10) (which is equivalent to the Cunningham number 10¹⁵⁷⁶⁸−1) and Φ₇₈₈₄(10) is an algebraic factor of the Cunningham number 10¹⁵⁷⁶⁸−1, the N+1 of "27227_3741" is 4 × "6805w7_3739", the N−1 of "68883_5132" is 62 × R₅₁₃₂(10) (which is equivalent to the Cunningham number 10⁵¹³²−1) and Φ₁₂₈₃(10) is an algebraic factor of the Cunningham number 10⁵¹³²−1, the N−1 of "79921_11629" is 720 × R₁₁₆₂₈(10) (which is equivalent to the Cunningham number 10¹¹⁶²⁸−1) and Φ₂₉₀₇(10) is an algebraic factor of the Cunningham number 10¹¹⁶²⁸−1, the N−1 of "80081_5736" is 80 × S₅₇₃₅(10) (which is equivalent to the Cunningham number 10⁵⁷³⁵+1) and Φ₁₁₄₇₀(10) is an algebraic factor of the Cunningham number 10⁵⁷³⁵+1, the N+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", etc.)

range of bases b	the factors of bⁿ±1 with 1 ≤ n ≤ 100	the factors of bⁿ±1 with 101 ≤ n ≤ 200	the factors of bⁿ±1 with 201 ≤ n ≤ 300	the factors of bⁿ±1 with 301 ≤ n ≤ 400	the first holes of bⁿ±1 and their known prime factors
2 ≤ b ≤ 100	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=2&TBase=100&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=A (bases 13 ≤ b ≤ 100 instead of 2 ≤ b ≤ 100, also list known prime factors only for bⁿ < 10²⁵⁵), http://maths-people.anu.edu.au/~brent/ftp/factors/holes.txt (bases 13 ≤ b ≤ 100 instead of 2 ≤ b ≤ 100, also list known prime factors only for bⁿ < 10²⁵⁵)
101 ≤ b ≤ 200	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=101&TBase=200&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=B (bases 101 ≤ b ≤ 199 instead of 101 ≤ b ≤ 200, also list known prime factors only for bⁿ < 10²⁵⁵)
201 ≤ b ≤ 300	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=201&TBase=300&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=C (bases 200 ≤ b ≤ 299 instead of 201 ≤ b ≤ 300, also list known prime factors only for bⁿ < 10²⁵⁵)
301 ≤ b ≤ 400	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)	http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=301&TBase=400&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=D (bases 300 ≤ b ≤ 400 instead of 301 ≤ b ≤ 400, also list known prime factors only for bⁿ < 10²⁵⁵)

The Cunningham numbers bⁿ±1 has algebraic factorization to product of the cyclotomic numbers Φ_d(b) with positive integers d dividing n (the bⁿ−1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization) or positive integers d dividing 2×n but not dividing n (the bⁿ+1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf) and 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) and http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf))

The Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf)) for the the cyclotomic numbers Φ_d(b) for bases 2 ≤ b ≤ 36 are: (for more information, see https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_list and http://myfactorcollection.mooo.com:8090/LCD_2_199 and http://myfactorcollection.mooo.com:8090/LCD_2_998 and https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm and https://en.wikipedia.org/wiki/Aurifeuillean_factorization#Examples (see the table) and https://www.rieselprime.de/ziki/Aurifeuillian_factor (see the table))

(Note: although there are no OEIS sequences of the Aurifeuillean factors of bⁿ±1 for bases b > 12, but there are OEIS sequences of the Aurifeuillean factors of p^p±1 for prime bases p, for primes p == 1 mod 4, p^p−1 has Aurifeuillean factors, and the OEIS sequence of the Aurifeuillean L factor is https://oeis.org/A352711, and the OEIS sequence of the Aurifeuillean M factor is https://oeis.org/A352732, for primes p == 3 mod 4, p^p+1 has Aurifeuillean factors, and the OEIS sequence of the Aurifeuillean L factor is https://oeis.org/A352400, and the OEIS sequence of the Aurifeuillean M factor is https://oeis.org/A352401)

b	the algebraic factor of bⁿ±1 which has Aurifeuillean factorization	the Aurifeuillean L factor	the Aurifeuillean M factor	examples of the Aurifeuillean factorization for exponent 2×r+1 = 37 (see the "Aurifeuillian" section)	OEIS sequence of the Aurifeuillean L factor	OEIS sequence of the Aurifeuillean M factor
2, 4, 8, 16, 32	Φ₄(2^2×r+1)	2^2×r+1−2^r+1+1	2^2×r+1+2^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=2&Exp=74&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A092440	https://oeis.org/A085601
3, 9, 27	Φ₆(3^2×r+1)	3^2×r+1−3^r+1+1	3^2×r+1+3^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=111&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220978	https://oeis.org/A198410
5, 25	Φ₅(5^2×r+1)	5^4×r+2−5^3×r+2+3×5^2×r+1−5^r+1+1	5^4×r+2+5^3×r+2+3×5^2×r+1+5^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=5&Exp=185&LBIDPMList=A&LBIDLODList=D	https://oeis.org/A220979	https://oeis.org/A220980
6, 36	Φ₁₂(6^2×r+1)	6^4×r+2−6^3×r+2+3×6^2×r+1−6^r+1+1	6^4×r+2+6^3×r+2+3×6^2×r+1+6^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=6&Exp=222&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220981	https://oeis.org/A220982
7	Φ₁₄(7^2×r+1)	7^6×r+3−7^5×r+3+3×7^4×r+2−7^3×r+2+3×7^2×r+1−7^r+1+1	7^6×r+3+7^5×r+3+3×7^4×r+2+7^3×r+2+3×7^2×r+1+7^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=7&Exp=259&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220983	https://oeis.org/A220984
10	Φ₂₀(10^2×r+1)	10^8×r+4−10^7×r+4+5×10^6×r+3−2×10^5×r+3+7×10^4×r+2−2×10^3×r+2+5×10^2×r+1−10^r+1+1	10^8×r+4+10^7×r+4+5×10^6×r+3+2×10^5×r+3+7×10^4×r+2+2×10^3×r+2+5×10^2×r+1+10^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=10&Exp=370&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220985	https://oeis.org/A220986
11	Φ₂₂(11^2×r+1)	11^10×r+5−11^9×r+5+5×11^8×r+4−11^7×r+4−11^6×r+3+11^5×r+3−11^4×r+2−11^3×r+2+5×11^2×r+1−11^r+1+1	11^10×r+5+11^9×r+5+5×11^8×r+4+11^7×r+4−11^6×r+3−11^5×r+3−11^4×r+2+11^3×r+2+5×11^2×r+1+11^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=11&Exp=407&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220987	https://oeis.org/A220988
12	Φ₆(12^2×r+1)	12^2×r+1−6×12^r+1	12^2×r+1+6×12^r+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=12&Exp=111&LBIDPMList=B&LBIDLODList=D	https://oeis.org/A220989	https://oeis.org/A220990
13	Φ₁₃(13^2×r+1)	13^12×r+6−13^11×r+6+7×13^10×r+5−3×13^9×r+5+15×13^8×r+4−5×13^7×r+4+19×13^6×r+3−5×13^5×r+3+15×13^4×r+2−3×13^3×r+2+7×13^2×r+1−13^r+1+1	13^12×r+6+13^11×r+6+7×13^10×r+5+3×13^9×r+5+15×13^8×r+4+5×13^7×r+4+19×13^6×r+3+5×13^5×r+3+15×13^4×r+2+3×13^3×r+2+7×13^2×r+1+13^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=481&LBIDPMList=A&LBIDLODList=D	–	–
14	Φ₂₈(14^2×r+1)	14^12×r+6−14^11×r+6+7×14^10×r+5−2×14^9×r+5+3×14^8×r+4+14^7×r+4−7×14^6×r+3+14^5×r+3+3×14^4×r+2−2×14^3×r+2+7×14^2×r+1−14^r+1+1	14^12×r+6+14^11×r+6+7×14^10×r+5+2×14^9×r+5+3×14^8×r+4-14^7×r+4-7×14^6×r+3-14^5×r+3+3×14^4×r+2+2×14^3×r+2+7×14^2×r+1+14^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=14&Exp=518&LBIDPMList=B&LBIDLODList=D	–	–
15	Φ₃₀(15^2×r+1)	15^8×r+4−15^7×r+4+8×15^6×r+3−3×15^5×r+3+13×15^4×r+2−3×15^3×r+2+8×15^2×r+1−15^r+1+1	15^8×r+4+15^7×r+4+8×15^6×r+3+3×15^5×r+3+13×15^4×r+2+3×15^3×r+2+8×15^2×r+1+15^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=15&Exp=555&LBIDPMList=B&LBIDLODList=D	–	–
17	Φ₁₇(17^2×r+1)	17^16×r+8−17^15×r+8+9×17^14×r+7−3×17^13×r+7+11×17^12×r+6−17^11×r+6−5×17^10×r+5+3×17^9×r+5−15×17^8×r+4+3×17^7×r+4−5×17^6×r+3−17^5×r+3+11×17^4×r+2−3×17^3×r+2+9×17^2×r+1−17^r+1+1	17^16×r+8+17^15×r+8+9×17^14×r+7+3×17^13×r+7+11×17^12×r+6+17^11×r+6−5×17^10×r+5−3×17^9×r+5−15×17^8×r+4−3×17^7×r+4−5×17^6×r+3+17^5×r+3+11×17^4×r+2+3×17^3×r+2+9×17^2×r+1+17^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=629&LBIDPMList=A&LBIDLODList=D	–	–
18	Φ₄(18^2×r+1)	18^2×r+1−6×18^r+1	18^2×r+1+6×18^r+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=18&Exp=74&LBIDPMList=B&LBIDLODList=D	–	–
19	Φ₃₈(19^2×r+1)	19^18×r+9−19^17×r+9+9×19^16×r+8−3×19^15×r+8+17×19^14×r+7−5×19^13×r+7+27×19^12×r+6−7×19^11×r+6+31×19^10×r+5−7×19^9×r+5+31×19^8×r+4−7×19^7×r+4+27×19^6×r+3−5×19^5×r+3+17×19^4×r+2−3×19^3×r+2+9×19^2×r+1−19^r+1+1	19^18×r+9+19^17×r+9+9×19^16×r+8+3×19^15×r+8+17×19^14×r+7+5×19^13×r+7+27×19^12×r+6+7×19^11×r+6+31×19^10×r+5+7×19^9×r+5+31×19^8×r+4+7×19^7×r+4+27×19^6×r+3+5×19^5×r+3+17×19^4×r+2+3×19^3×r+2+9×19^2×r+1+19^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=19&Exp=703&LBIDPMList=B&LBIDLODList=D	–	–
20	Φ₅(20^2×r+1)	20^4×r+2−10×20^3×r+1+3×20^2×r+1−10×20^r+1	20^4×r+2+10×20^3×r+1+3×20^2×r+1+10×20^r+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=20&Exp=185&LBIDPMList=A&LBIDLODList=D	–	–
21	Φ₂₁(21^2×r+1)	21^12×r+6−21^11×r+6+10×21^10×r+5−3×21^9×r+5+13×21^8×r+4−2×21^7×r+4+7×21^6×r+3−2×21^5×r+3+13×21^4×r+2−3×21^3×r+2+10×21^2×r+1−21^r+1+1	21^12×r+6+21^11×r+6+10×21^10×r+5+3×21^9×r+5+13×21^8×r+4+2×21^7×r+4+7×21^6×r+3+2×21^5×r+3+13×21^4×r+2+3×21^3×r+2+10×21^2×r+1+21^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=777&LBIDPMList=A&LBIDLODList=D	–	–
22	Φ₄₄(22^2×r+1)	22^20×r+10−22^19×r+10+11×22^18×r+9−4×22^17×r+9+27×22^16×r+8−7×22^15×r+8+33×22^14×r+7−6×22^13×r+7+21×22^12×r+6−3×22^11×r+6+11×22^10×r+5−3×22^9×r+5+21×22^8×r+4−6×22^7×r+4+33×22^6×r+3−7×22^5×r+3+27×22^4×r+2−4×22^3×r+2+11×22^2×r+1−22^r+1+1	22^20×r+10+22^19×r+10+11×22^18×r+9+4×22^17×r+9+27×22^16×r+8+7×22^15×r+8+33×22^14×r+7+6×22^13×r+7+21×22^12×r+6+3×22^11×r+6+11×22^10×r+5+3×22^9×r+5+21×22^8×r+4+6×22^7×r+4+33×22^6×r+3+7×22^5×r+3+27×22^4×r+2+4×22^3×r+2+11×22^2×r+1+22^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=22&Exp=814&LBIDPMList=B&LBIDLODList=D	–	–
23	Φ₄₆(23^2×r+1)	23^22×r+11−23^21×r+11+11×23^20×r+10−3×23^19×r+10+9×23^18×r+9+23^17×r+9−19×23^16×r+8+5×23^15×r+8−15×23^14×r+7−23^13×r+7+25×23^12×r+6−7×23^11×r+6+25×23^10×r+5−23^9×r+5−15×23^8×r+4+5×23^7×r+4−19×23^6×r+3+23^5×r+3+9×23^4×r+2−3×23^3×r+2+11×23^2×r+1−23^r+1+1	23^22×r+11+23^21×r+11+11×23^20×r+10+3×23^19×r+10+9×23^18×r+9−23^17×r+9−19×23^16×r+8−5×23^15×r+8−15×23^14×r+7+23^13×r+7+25×23^12×r+6+7×23^11×r+6+25×23^10×r+5+23^9×r+5−15×23^8×r+4−5×23^7×r+4−19×23^6×r+3−23^5×r+3+9×23^4×r+2+3×23^3×r+2+11×23^2×r+1+23^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=23&Exp=851&LBIDPMList=B&LBIDLODList=D	–	–
24	Φ₁₂(24^2×r+1)	24^4×r+2−12×24^3×r+1+3×24^2×r+1−12×24^r+1	24^4×r+2+12×24^3×r+1+3×24^2×r+1+12×24^r+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=24&Exp=222&LBIDPMList=B&LBIDLODList=D	–	–
26	Φ₅₂(26^2×r+1)	26^24×r+12−26^23×r+12+13×26^22×r+11−4×26^21×r+11+19×26^20×r+10−26^19×r+10−13×26^18×r+9+4×26^17×r+9−11×26^16×r+8−26^15×r+8+13×26^14×r+7−2×26^13×r+7+7×26^12×r+6−2×26^11×r+6+13×26^10×r+5−26^9×r+5−11×26^8×r+4+4×26^7×r+4−13×26^6×r+3−26^5×r+3+19×26^4×r+2−4×26^3×r+2+13×26^2×r+1−26^r+1+1	26^24×r+12+26^23×r+12+13×26^22×r+11+4×26^21×r+11+19×26^20×r+10+26^19×r+10−13×26^18×r+9−4×26^17×r+9−11×26^16×r+8+26^15×r+8+13×26^14×r+7+2×26^13×r+7+7×26^12×r+6+2×26^11×r+6+13×26^10×r+5+26^9×r+5−11×26^8×r+4−4×26^7×r+4−13×26^6×r+3+26^5×r+3+19×26^4×r+2+4×26^3×r+2+13×26^2×r+1+26^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=962&LBIDPMList=B&LBIDLODList=D	–	–
28	Φ₁₄(28^2×r+1)	28^6×r+3−14×28^5×r+2+3×28^4×r+2−14×28^3×r+1+3×28^2×r+1−14×28^r+1	28^6×r+3+14×28^5×r+2+3×28^4×r+2+14×28^3×r+1+3×28^2×r+1+14×28^r+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=28&Exp=259&LBIDPMList=B&LBIDLODList=D	–	–
29	Φ₂₉(29^2×r+1)	29^28×r+14−29^27×r+14+15×29^26×r+13−5×29^25×r+13+33×29^24×r+12−5×29^23×r+12+13×29^22×r+11−29^21×r+11+15×29^20×r+10−7×29^19×r+10+57×29^18×r+9−11×29^17×r+9+45×29^16×r+8−5×29^15×r+8+19×29^14×r+7−5×29^13×r+7+45×29^12×r+6−11×29^11×r+6+57×29^10×r+5−7×29^9×r+5+15×29^8×r+4−29^7×r+4+13×29^6×r+3−5×29^5×r+3+33×29^4×r+2−5×29^3×r+2+15×29^2×r+1−29^r+1+1	29^28×r+14+29^27×r+14+15×29^26×r+13+5×29^25×r+13+33×29^24×r+12+5×29^23×r+12+13×29^22×r+11+29^21×r+11+15×29^20×r+10+7×29^19×r+10+57×29^18×r+9+11×29^17×r+9+45×29^16×r+8+5×29^15×r+8+19×29^14×r+7+5×29^13×r+7+45×29^12×r+6+11×29^11×r+6+57×29^10×r+5+7×29^9×r+5+15×29^8×r+4+29^7×r+4+13×29^6×r+3+5×29^5×r+3+33×29^4×r+2+5×29^3×r+2+15×29^2×r+1+29^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=29&Exp=1073&LBIDPMList=A&LBIDLODList=D	–	–
30	Φ₆₀(30^2×r+1)	30^16×r+8−30^15×r+8+15×30^14×r+7−5×30^13×r+7+38×30^12×r+6−8×30^11×r+6+45×30^10×r+5−8×30^9×r+5+43×30^8×r+4−8×30^7×r+4+45×30^6×r+3−8×30^5×r+3+38×30^4×r+2−5×30^3×r+2+15×30^2×r+1−30^r+1+1	30^16×r+8+30^15×r+8+15×30^14×r+7+5×30^13×r+7+38×30^12×r+6+8×30^11×r+6+45×30^10×r+5+8×30^9×r+5+43×30^8×r+4+8×30^7×r+4+45×30^6×r+3+8×30^5×r+3+38×30^4×r+2+5×30^3×r+2+15×30^2×r+1+30^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=30&Exp=1110&LBIDPMList=B&LBIDLODList=D	–	–
31	Φ₆₂(31^2×r+1)	31^30×r+15−31^29×r+15+15×31^28×r+14−5×31^27×r+14+43×31^26×r+13−11×31^25×r+13+83×31^24×r+12−19×31^23×r+12+125×31^22×r+11−25×31^21×r+11+151×31^20×r+10−29×31^19×r+10+169×31^18×r+9−31×31^17×r+9+173×31^16×r+8−31×31^15×r+8+173×31^14×r+7−31×31^13×r+7+169×31^12×r+6−29×31^11×r+6+151×31^10×r+5−25×31^9×r+5+125×31^8×r+4−19×31^7×r+4+83×31^6×r+3−11×31^5×r+3+43×31^4×r+2−5×31^3×r+2+15×31^2×r+1−31^r+1+1	31^30×r+15+31^29×r+15+15×31^28×r+14+5×31^27×r+14+43×31^26×r+13+11×31^25×r+13+83×31^24×r+12+19×31^23×r+12+125×31^22×r+11+25×31^21×r+11+151×31^20×r+10+29×31^19×r+10+169×31^18×r+9+31×31^17×r+9+173×31^16×r+8+31×31^15×r+8+173×31^14×r+7+31×31^13×r+7+169×31^12×r+6+29×31^11×r+6+151×31^10×r+5+25×31^9×r+5+125×31^8×r+4+19×31^7×r+4+83×31^6×r+3+11×31^5×r+3+43×31^4×r+2+5×31^3×r+2+15×31^2×r+1+31^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=31&Exp=1147&LBIDPMList=B&LBIDLODList=D	–	–
33	Φ₃₃(33^2×r+1)	33^20×r+10−33^19×r+10+16×33^18×r+9−5×33^17×r+9+37×33^16×r+8−6×33^15×r+8+19×33^14×r+7+33^13×r+7−32×33^12×r+6+9×33^11×r+6−59×33^10×r+5+9×33^9×r+5−32×33^8×r+4+33^7×r+4+19×33^6×r+3−6×33^5×r+3+37×33^4×r+2−5×33^3×r+2+16×33^2×r+1−33^r+1+1	33^20×r+10+33^19×r+10+16×33^18×r+9+5×33^17×r+9+37×33^16×r+8+6×33^15×r+8+19×33^14×r+7−33^13×r+7−32×33^12×r+6−9×33^11×r+6−59×33^10×r+5−9×33^9×r+5−32×33^8×r+4−33^7×r+4+19×33^6×r+3+6×33^5×r+3+37×33^4×r+2+5×33^3×r+2+16×33^2×r+1+33^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=33&Exp=1221&LBIDPMList=A&LBIDLODList=D	–	–
34	Φ₆₈(34^2×r+1)	34^32×r+16−34^31×r+16+17×34^30×r+15−6×34^29×r+15+59×34^28×r+14−15×34^27×r+14+119×34^26×r+13−26×34^25×r+13+181×34^24×r+12−35×34^23×r+12+221×34^22×r+11−40×34^21×r+11+243×34^20×r+10−43×34^19×r+10+255×34^18×r+9−44×34^17×r+9+257×34^16×r+8−44×34^15×r+8+255×34^14×r+7−43×34^13×r+7+243×34^12×r+6−40×34^11×r+6+221×34^10×r+5−35×34^9×r+5+181×34^8×r+4−26×34^7×r+4+119×34^6×r+3−15×34^5×r+3+59×34^4×r+2−6×34^3×r+2+17×34^2×r+1−34^r+1+1	34^32×r+16+34^31×r+16+17×34^30×r+15+6×34^29×r+15+59×34^28×r+14+15×34^27×r+14+119×34^26×r+13+26×34^25×r+13+181×34^24×r+12+35×34^23×r+12+221×34^22×r+11+40×34^21×r+11+243×34^20×r+10+43×34^19×r+10+255×34^18×r+9+44×34^17×r+9+257×34^16×r+8+44×34^15×r+8+255×34^14×r+7+43×34^13×r+7+243×34^12×r+6+40×34^11×r+6+221×34^10×r+5+35×34^9×r+5+181×34^8×r+4+26×34^7×r+4+119×34^6×r+3+15×34^5×r+3+59×34^4×r+2+6×34^3×r+2+17×34^2×r+1+34^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=34&Exp=1258&LBIDPMList=B&LBIDLODList=D	–	–
35	Φ₇₀(35^2×r+1)	35^24×r+12−35^23×r+12+18×35^22×r+11−6×35^21×r+11+48×35^20×r+10−7×35^19×r+10+11×35^18×r+9+5×35^17×r+9−55×35^16×r+8+8×35^15×r+8−11×35^14×r+7−5×35^13×r+7+47×35^12×r+6−5×35^11×r+6−11×35^10×r+5+8×35^9×r+5−55×35^8×r+4+5×35^7×r+4+11×35^6×r+3−7×35^5×r+3+48×35^4×r+2−6×35^3×r+2+18×35^2×r+1−35^r+1+1	35^24×r+12+35^23×r+12+18×35^22×r+11+6×35^21×r+11+48×35^20×r+10+7×35^19×r+10+11×35^18×r+9−5×35^17×r+9−55×35^16×r+8−8×35^15×r+8−11×35^14×r+7+5×35^13×r+7+47×35^12×r+6+5×35^11×r+6−11×35^10×r+5−8×35^9×r+5−55×35^8×r+4−5×35^7×r+4+11×35^6×r+3+7×35^5×r+3+48×35^4×r+2+6×35^3×r+2+18×35^2×r+1+35^r+1+1	http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=35&Exp=1295&LBIDPMList=B&LBIDLODList=D	–	–

If core(b) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of b) is == 1 mod 4 (and core(b) > 1, i.e. b is not a square number (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290)), then bⁿ−1 has Aurifeuillean factorization if and only if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)) (and bⁿ+1 has no Aurifeuillean factorization for any n), and if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)), we write Φ_nL(b) = gcd(Φ_n(b), the Aurifeuillean L factor of Φ_core(b)(b^n/core(b))) and Φ_nM(b) = gcd(Φ_n(b), the Aurifeuillean M factor of Φ_core(b)(b^n/core(b))); if core(b) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of b) is == 2, 3 mod 4, then bⁿ+1 has Aurifeuillean factorization if and only if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)) (and bⁿ−1 has no Aurifeuillean factorization for any n), and if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)), we write Φ_(2×n)L(b) = gcd(Φ_2×n(b), the Aurifeuillean L factor of Φ_2×core(b)(b^n/core(b))) and Φ_(2×n)M(b) = gcd(Φ_2×n(b), the Aurifeuillean M factor of Φ_2×core(b)(b^n/core(b))). Thus, Φ_nL(b) is "L'_n" (if core(b) == 1 mod 4) or "L'_n/2" (if core(b) == 2, 3 mod 4) in the section "Algebraic Factors for aⁿ±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "a" instead of "b" for the base), and Φ_nM(b) is "M'_n" (if core(b) == 1 mod 4) or "M'_n/2" (if core(b) == 2, 3 mod 4) in the section "Algebraic Factors for aⁿ±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "a" instead of "b" for the base), for the example of base b = 10 see https://stdkmd.net/nrr/repunit/phin10.htm (the numbers Φ_20L(10), Φ_20M(10), Φ_60L(10), Φ_60M(10), Φ_100L(10), Φ_100M(10), Φ_140L(10), Φ_140M(10), Φ_180L(10), Φ_180M(10), ...) and https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_repunit.

the 3168th minimal prime in base 13, 9₃₀₈1, N−1 is 117×R₃₀₈(13), thus factor N−1 is equivalent to factor the Cunningham number 13³⁰⁸−1, and for the algebraic factors of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=308&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=&LM=
the 3179th minimal prime in base 13, B₅₆₃C, N−1 is 11×R₅₆₄(13), thus factor N−1 is equivalent to factor the Cunningham number 13⁵⁶⁴−1, and for the algebraic factors of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=564&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=&LM=
the 3180th minimal prime in base 13, 1B₅₇₆, N−1 is 23×R₅₇₆(13), thus factor N−1 is equivalent to factor the Cunningham number 13⁵⁷⁶−1, and for the algebraic factors of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=576&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=&LM=
the 10320th minimal prime in base 17, 9₂₉₂1, N−1 is 153×R₂₉₂(17), thus factor N−1 is equivalent to factor the Cunningham number 17²⁹²−1, and for the algebraic factors of 17²⁹²−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=292&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 17²⁹²−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=292&c0=-&EN=&LM=
the 13304th minimal prime in base 21, 7₂₃₀1, N−1 is 147×R₂₃₀(21), thus factor N−1 is equivalent to factor the Cunningham number 21²³⁰−1, and for the algebraic factors of 21²³⁰−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=230&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 21²³⁰−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=230&c0=-&EN=&LM=
the 13355th minimal prime in base 21, 3₁₀₆₃2, N+1 is 3×R₁₀₆₄(21), thus factor N−1 is equivalent to factor the Cunningham number 21¹⁰⁶⁴−1, and for the algebraic factors of 21¹⁰⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=1064&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 21¹⁰⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=1064&c0=-&EN=&LM=
the 25199th minimal prime in base 26, 9K₃₄₃AP, N+1 is 6370×R₃₄₄(26), thus factor N+1 is equivalent to factor the Cunningham number 26³⁴⁴−1, and for the algebraic factors of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=344&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=&LM=
the 25200th minimal prime in base 26, 8₃₅₄1, N−1 is 208×R₃₅₄(26), thus factor N−1 is equivalent to factor the Cunningham number 26³⁵⁴−1, and for the algebraic factors of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=354&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=&LM=

The helper file for the 3168th minimal prime in base 13 (9₃₀₈1) in factordb: http://factordb.com/helper.php?id=1100000000840126705

The helper file for the 3179th minimal prime in base 13 (B₅₆₃C) in factordb: http://factordb.com/helper.php?id=1100000000000217927

The helper file for the 3180th minimal prime in base 13 (1B₅₇₆) in factordb: http://factordb.com/helper.php?id=1100000002321021456

The helper file for the 10320th minimal prime in base 17 (9₂₉₂1) in factordb: http://factordb.com/helper.php?id=1100000000840355814

The helper file for the 13304th minimal prime in base 21 (7₂₃₀1) in factordb: http://factordb.com/helper.php?id=1100000002325398836

The helper file for the 13355th minimal prime in base 21 (3₁₀₆₃2) in factordb: http://factordb.com/helper.php?id=1100000002325396014

The helper file for the 25199th minimal prime in base 26 (9K₃₄₃AP) in factordb: http://factordb.com/helper.php?id=1100000000840632228

The helper file for the 25200th minimal prime in base 26 (8₃₅₄1) in factordb: http://factordb.com/helper.php?id=1100000000840632517

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3168th minimal prime in base 13 (9₃₀₈1) in factordb: http://factordb.com/index.php?id=1100000000840126706&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3179th minimal prime in base 13 (B₅₆₃C) in factordb: http://factordb.com/index.php?id=1100000000271764311&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3180th minimal prime in base 13 (1B₅₇₆) in factordb: http://factordb.com/index.php?id=1100000002321021531&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 10320th minimal prime in base 17 (9₂₉₂1) in factordb: http://factordb.com/index.php?id=1100000000840355817&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 13304th minimal prime in base 21 (7₂₃₀1) in factordb: http://factordb.com/index.php?id=1100000002325398854&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 13355th minimal prime in base 21 (3₁₀₆₃2) in factordb: http://factordb.com/index.php?id=1100000002325396028&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 25199th minimal prime in base 26 (9K₃₄₃AP) in factordb: http://factordb.com/index.php?id=1100000000840632232&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 25200th minimal prime in base 26 (8₃₅₄1) in factordb: http://factordb.com/index.php?id=1100000000840632623&open=ecm

(in the tables below, Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240))

(for the prime factors > 10²⁴ (other than the ultimate prime factor (https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):")) of each algebraic factor) in the tables below, "ECM" means 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://www.numericana.com/answer/factoring.htm#ecm, http://factordb.com/listecm.php?c=1, http://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, http://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html), "P−1" means 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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php), "P+1" means 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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php), "SNFS" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/), "GNFS" means the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/))

For the number 13³⁰⁸−1, it is the product of Φ_d(13) with positive integers d dividing 308 (i.e. d = 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
Φ₁(13)	2² × 3
Φ₂(13)	2 × 7
Φ₄(13)	2 × 5 × 17
Φ₇(13)	5229043
Φ₁₁(13)	23 × 419 × 859 × 18041
Φ₁₄(13)	7 × 29 × 22079
Φ₂₂(13)	128011456717
Φ₂₈(13)	23161037562937
Φ₄₄(13)	5281 × 3577574298489429481
Φ₇₇(13)	624958606550654822293 × (47-digit prime)
Φ₁₅₄(13)	78947177 × (59-digit prime)
Φ₃₀₈(13)	7393 × 1702933 × 150324329 × 718377597171850001 × 4209006442599882158485591696242263069 × (61-digit prime)

For the number 13⁵⁶⁴−1, it is the product of Φ_d(13) with positive integers d dividing 564 (i.e. d = 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
Φ₁(13)	2² × 3
Φ₂(13)	2 × 7
Φ₃(13)	3 × 61
Φ₄(13)	2 × 5 × 17
Φ₆(13)	157
Φ₁₂(13)	28393
Φ₄₇(13)	183959 × 19216136497 × 534280344481909234853671069326391741
Φ₉₄(13)	498851139881 × 3245178229485124818467952891417691434077
Φ₁₄₁(13)	283 × 1693 × 1924651 × 455036140638637 × (76-digit prime)
Φ₁₈₈(13)	36097 × 75389 × 99886248944632632917 × (74-digit prime)
Φ₂₈₂(13)	590202369266263393 × (85-digit prime)
Φ₅₆₄(13)	233628485003849577181 × 94531330515097101267386264339794253977 (ECM (Montgomery curve), B1 = 3000000, Sigma = 2146847123, the prime factorization of the group order is 2³ × 3³ × 5 × 11 × 23 × 4871 × 10099 × 17207 × 1389277 × 2661643 × 110532803) × 27969827431131578608318126024627616357147784803797 (GNFS) × (98-digit prime)

For the number 13⁵⁷⁶−1, it is the product of Φ_d(13) with positive integers d dividing 576 (i.e. d = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
Φ₁(13)	2² × 3
Φ₂(13)	2 × 7
Φ₃(13)	3 × 61
Φ₄(13)	2 × 5 × 17
Φ₆(13)	157
Φ₈(13)	2 × 14281
Φ₉(13)	3 × 1609669
Φ₁₂(13)	28393
Φ₁₆(13)	2 × 407865361
Φ₁₈(13)	19 × 271 × 937
Φ₂₄(13)	815702161
Φ₃₂(13)	2 × 2657 × 441281 × 283763713
Φ₃₆(13)	37 × 428041 × 1471069
Φ₄₈(13)	1009 × 659481276875569
Φ₆₄(13)	2 × 193 × 1601 × 10433 × 68675120456139881482562689
Φ₇₂(13)	73 × 4177 × 181297 × 9818892432332713
Φ₉₆(13)	97 × 88993 × 127028743393 × 403791981344275297
Φ₁₄₄(13)	3889 × 680401 × 29975087953 × 6654909974864689 × 558181416418089697
Φ₁₉₂(13)	1153 × 11352931040252580224415980746369 × 14977427998321433931503086910333672833
Φ₂₈₈(13)	2017 × 47521 × 54721 × 1590049 × 8299042833797200969471889569 × (61-digit prime)
Φ₅₇₆(13)	577 × 6337 × 5247817273269739636080024961 × 5497355933986265726220616321 × 1032606621363411464640473542092061600217962755283816476128113983937 (GNFS) × (86-digit prime)

For the number 17²⁹²−1, it is the product of Φ_d(17) with positive integers d dividing 292 (i.e. d = 1, 2, 4, 73, 146, 292), and the factorization of Φ_d(17) for these positive integers d are:

from	prime factorization
Φ₁(17)	2⁴
Φ₂(17)	2 × 3²
Φ₄(17)	2 × 5 × 29
Φ₇₃(17)	293 × 1621745371 × 3038535503 × 319344640907 × 596137412912777 × (42-digit prime)
Φ₁₄₆(17)	284117 × 1517302254487813 × (68-digit prime)
Φ₂₉₂(17)	877 × 4673 × 734854949942641407221086529797643582883477575817985261656573824687508604116290261093 × (87-digit prime)

For the number 21²³⁰−1, it is the product of Φ_d(21) with positive integers d dividing 230 (i.e. d = 1, 2, 5, 10, 23, 46, 115, 230), and the factorization of Φ_d(21) for these positive integers d are:

from	prime factorization
Φ₁(21)	2² × 5
Φ₂(21)	2 × 11
Φ₅(21)	5 × 40841
Φ₁₀(21)	185641
Φ₂₃(21)	47 × 19597 × 139870566115103282847737
Φ₄₆(21)	277 × 461 × 599 × 691 × 2215825387044753577
Φ₁₁₅(21)	1381 × 282924347471791 × 3394964812534556016503466037951 × (69-digit prime)
Φ₂₃₀(21)	2531 × 11731 × 22952851 × 595377311 × 688660481 × 58286351831 × 69727564981 × (63-digit prime)

For the number 21¹⁰⁶⁴−1, it is the product of Φ_d(21) with positive integers d dividing 1064 (i.e. d = 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532, 1064), and the factorization of Φ_d(21) for these positive integers d are:

from	currently known prime factorization
Φ₁(21)	2² × 5
Φ₂(21)	2 × 11
Φ₄(21)	2 × 13 × 17
Φ₇(21)	43 × 631 × 3319
Φ₈(21)	2 × 97241
Φ₁₄(21)	81867661
Φ₁₉(21)	12061389013 × 54921106624003
Φ₂₈(21)	29 × 3697 × 68454248717
Φ₃₈(21)	609673 × 987749814642143197
Φ₅₆(21)	617 × 912521 × 115593326297 × 831380909129
Φ₇₆(21)	229 × 457 × (43-digit prime)
Φ₁₃₃(21)	948175293266954869500463698756935713088089028515629708586399 × (83-digit prime)
Φ₁₅₂(21)	136649 × 6629177 × 8871582886760161 × 4370570172021545617284038736601 × 4510053597010461591911520110711387257
Φ₂₆₆(21)	4523 × 263478423344974307 × 39188712102054729290763779 × 1027619231425962708522784338595411210117 × (58-digit prime)
Φ₅₃₂(21)	1080514246723801 × 4598307023923376056176577 (P−1, B1 = 100000, B2 = 39772318, the prime factorization of P−1 is 2⁶ × 3 × 7 × 11 × 19 × 23 × 241 × 1229 × 1697 × 7369 × 192161) × 173111326443349916878938361 × (220-digit composite with no known proper factor, SNFS difficulty is 301.466, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=21&Exp=266&c0=%2B&LM=&SA=, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%2821%5E266%2B1%29442%2F%2821%5E38%2B1%29%2F%2821%5E14%2B1%29&action=all&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)
Φ₁₀₆₄(21)	140449 × 723460417 × (558-digit composite with no known proper factor, SNFS difficulty is 602.932, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=21&Exp=532&c0=%2B&LM=&SA=)

For the number 26³⁴⁴−1, it is the product of Φ_d(26) with positive integers d dividing 344 (i.e. d = 1, 2, 4, 8, 43, 86, 172, 344), and the factorization of Φ_d(26) for these positive integers d are:

from	prime factorization
Φ₁(26)	5²
Φ₂(26)	3³
Φ₄(26)	677
Φ₈(26)	17 × 26881
Φ₄₃(26)	(60-digit prime)
Φ₈₆(26)	681293 × (54-digit prime)
Φ₁₇₂(26)	173 × 66221 × 97942133 × 338286119038330712762413 × 290239124722842089063959709049053 × (48-digit prime)
Φ₃₄₄(26)	259295161 × 14470172263033 × (217-digit prime)

For the number 26³⁵⁴−1, it is the product of Φ_d(26) with positive integers d dividing 354 (i.e. d = 1, 2, 3, 6, 59, 118, 177, 354), and the factorization of Φ_d(26) for these positive integers d are:

from	prime factorization
Φ₁(26)	5²
Φ₂(26)	3³
Φ₃(26)	19 × 37
Φ₆(26)	3 × 7 × 31
Φ₅₉(26)	3541 × 334945708538658924935948356996883525107 × 10265667109489266992108219345733472151257
Φ₁₁₈(26)	254250862891621 × (68-digit prime)
Φ₁₇₇(26)	47791 × 1311074895191091284466533625050044762267011115706300424823729 × (100-digit prime)
Φ₃₅₄(26)	709 × 16441898216641 × (149-digit prime)

the 151st minimal prime in base 9, 30₁₁₅₈11, N−1 is 9×S₂₃₁₉(3), thus factor N−1 is equivalent to factor the Cunningham number 3²³¹⁹+1, N−1 is only 12.693% factored (see http://factordb.com/index.php?id=1100000002376318423&open=prime), and for the algebraic factors of 3²³¹⁹+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=2319&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 3²³¹⁹+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=2319&c0=%2B&EN=&LM=
the 3187th minimal prime in base 13, 7₁₅₀₄1, N−1 is 91×R₁₅₀₄(13), thus factor N−1 is equivalent to factor the Cunningham number 13¹⁵⁰⁴−1, N−1 is only 28.604% factored (see http://factordb.com/index.php?id=1100000002320890755&open=prime) (since 28.604% is between 1/4 and 1/3, CHG proof is possible, however, since factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify CHG proofs, thus there is still primality certificate in factordb), and for the algebraic factors of 13¹⁵⁰⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=1504&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13¹⁵⁰⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1504&c0=-&EN=&LM=
the 2342nd minimal prime in base 16, 90₃₅₄₂91, N−1 is 144×S₃₅₄₃(16), thus factor N−1 is equivalent to factor the Cunningham number 16³⁵⁴³+1, N−1 is only 1.255% factored (see http://factordb.com/index.php?id=1100000000633424191&open=prime), and for the algebraic factors of 16³⁵⁴³+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=16&Exp=3543&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 16³⁵⁴³+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=16&Exp=3543&c0=%2B&EN=&LM=
the 10391st minimal prime in base 17, 1F₇₀₉₂, N−1 is 31×R₇₀₉₂(17), thus factor N−1 is equivalent to factor the Cunningham number 17⁷⁰⁹²−1, N−1 is only 7.085% factored (see http://factordb.com/index.php?id=1100000000840355927&open=prime), and for the algebraic factors of 17⁷⁰⁹²−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=7092&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 17⁷⁰⁹²−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=7092&c0=-&EN=&LM=
the 25240th minimal prime in base 26, 5₁₈₈₅4P, N+1 is 130×R₁₈₈₆(26), thus factor N+1 is equivalent to factor the Cunningham number 26¹⁸⁸⁶−1, N+1 is only 7.262% factored (see http://factordb.com/index.php?id=1100000003850155314&open=prime), and for the algebraic factors of 26¹⁸⁸⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=1886&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26¹⁸⁸⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=1886&c0=-&EN=&LM=
the 35277th minimal prime in base 36, OZ₃₉₃₂AZ, N+1 is 31500×R₃₉₃₃(36), thus factor N+1 is equivalent to factor the Cunningham number 36³⁹³³−1, N+1 is only 16.004% factored (see http://factordb.com/index.php?id=1100000000840634476&open=prime), and for the algebraic factors of 36³⁹³³−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=36&Exp=3933&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 36³⁹³³−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=36&Exp=3933&c0=-&EN=&LM=

The helper file for the 151st minimal prime in base 9 (30₁₁₅₈11) in factordb: http://factordb.com/helper.php?id=1100000002376318423

The helper file for the 3187th minimal prime in base 13 (7₁₅₀₄1) in factordb: http://factordb.com/helper.php?id=1100000002320890755

The helper file for the 2342nd minimal prime in base 16 (90₃₅₄₂91) in factordb: http://factordb.com/helper.php?id=1100000000633424191

The helper file for the 10391st minimal prime in base 17 (1F₇₀₉₂) in factordb: http://factordb.com/helper.php?id=1100000000840355927

The helper file for the 25240th minimal prime in base 26 (5₁₈₈₅4P) in factordb: http://factordb.com/helper.php?id=1100000003850155314

The helper file for the 35277th minimal prime in base 36 (OZ₃₉₃₂AZ) in factordb: http://factordb.com/helper.php?id=1100000000840634476

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 151st minimal prime in base 9 (30₁₁₅₈11) in factordb: http://factordb.com/index.php?id=1100000002376318436&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3187th minimal prime in base 13 (7₁₅₀₄1) in factordb: http://factordb.com/index.php?id=1100000002320890782&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 2342nd minimal prime in base 16 (90₃₅₄₂91) in factordb: http://factordb.com/index.php?id=1100000000633424203&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 10391st minimal prime in base 17 (1F₇₀₉₂) in factordb: http://factordb.com/index.php?id=1100000000840355928&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 25240th minimal prime in base 26 (5₁₈₈₅4P) in factordb: http://factordb.com/index.php?id=1100000003850159350&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 35277th minimal prime in base 36 (OZ₃₉₃₂AZ) in factordb: http://factordb.com/index.php?id=1100000000840634478&open=ecm

For the number 3²³¹⁹+1, it is the product of Φ_d(3) with positive integers d dividing 4638 but not dividing 2319 (i.e. d = 2, 6, 1546, 4638), and the factorization of Φ_d(3) for these positive integers d are: (since 6 and 4638 are == 6 mod 12, thus for these two positive integers d, Φ_d(3) has Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf)), and Φ_dL(3) and Φ_dM(3) are their Aurifeuillean L and M factors, respectively)

from	currently known prime factorization
Φ₂(3)	2²
Φ_6L(3)	1 (empty product (https://en.wikipedia.org/wiki/Empty_product))
Φ_6M(3)	7
Φ₁₅₄₆(3)	1182691 × 454333843 × 7175619780295897339 × 219067434459114063477547 × 650663511671253931884619 × (288-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=773&c0=%2B&LM=&SA=, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%283%5E773%2B1%29%2F4&action=all&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)
Φ_4638L(3)	18553 × 2957658597967379799686737984695290731543 × (325-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=L&SA=)
Φ_4638M(3)	4639 × 6716055901 × (356-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=M&SA=)

For the number 13¹⁵⁰⁴−1, it is the product of Φ_d(13) with positive integers d dividing 1504 (i.e. d = 1, 2, 4, 8, 16, 32, 47, 94, 188, 376, 752, 1504), and the factorization of Φ_d(13) for these positive integers d are:

from	currently known prime factorization
Φ₁(13)	2² × 3
Φ₂(13)	2 × 7
Φ₄(13)	2 × 5 × 17
Φ₈(13)	2 × 14281
Φ₁₆(13)	2 × 407865361
Φ₃₂(13)	2 × 2657 × 441281 × 283763713
Φ₄₇(13)	183959 × 19216136497 × 534280344481909234853671069326391741
Φ₉₄(13)	498851139881 × 3245178229485124818467952891417691434077
Φ₁₈₈(13)	36097 × 75389 × 99886248944632632917 × (74-digit prime)
Φ₃₇₆(13)	41737 × 553784729353 × 188172028979257 × 398225319299696783138113 × 7663511503164270157006126605793 × 8935170451146532986983277856738508374630999814576686938913 × (62-digit prime)
Φ₇₅₂(13)	13537 × 1232912541076129 × (391-digit composite with no known proper factor, SNFS difficulty is 421.071, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=376&c0=%2B&LM=&SA=)
Φ₁₅₀₄(13)	4513 × 9426289921 × (807-digit composite with no known proper factor, SNFS difficulty is 837.685, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=752&c0=%2B&LM=&SA=)

For the number 16³⁵⁴³+1 = 2¹⁴¹⁷²+1, it is the product of Φ_d(2) with positive integers d dividing 28344 but not dividing 14172 (i.e. d = 8, 24, 9448, 28344), and the factorization of Φ_d(2) for these positive integers d are:

from	currently known prime factorization
Φ₈(2)	17
Φ₂₄(2)	241
Φ₉₄₄₈(2)	107083633 × 7076306353 × 2428629073416562046689 × (1382-digit composite with no known proper factor, SNFS difficulty is 1422.066, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=4724&c0=%2B&LM=&SA=)
Φ₂₈₃₄₄(2)	265073089 × (2834-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=14172&c0=%2B&LM=&SA=)

For the number 17⁷⁰⁹²−1, it is the product of Φ_d(17) with positive integers d dividing 7092 (i.e. d = 1, 2, 3, 4, 6, 9, 12, 18, 36, 197, 394, 591, 788, 1182, 1773, 2364, 3546, 7092), and the factorization of Φ_d(17) for these positive integers d are:

from	currently known prime factorization
Φ₁(17)	2⁴
Φ₂(17)	2 × 3²
Φ₃(17)	307
Φ₄(17)	2 × 5 × 29
Φ₆(17)	3 × 7 × 13
Φ₉(17)	19 × 1270657
Φ₁₂(17)	83233
Φ₁₈(17)	3 × 1423 × 5653
Φ₃₆(17)	37 × 109 × 181 × 2089 × 382069
Φ₁₉₇(17)	646477768184104922935115731396719622668746018369021 × (191-digit prime)
Φ₃₉₄(17)	1720812337 × 120652139803422836046398107883 × 11854861245452004511262968204651829313761 × 930821833870171289422620828584179333038475130149 × (115-digit prime)
Φ₅₉₁(17)	150824383 × (475-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=-&LM=&SA=)
Φ₇₈₈(17)	(483-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=394&c0=%2B&LM=&SA=)
Φ₁₁₈₂(17)	3547 × 1924297 × (473-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=%2B&LM=&SA=)
Φ₁₇₇₃(17)	99289 × (1443-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=-&LM=&SA=)
Φ₂₃₆₄(17)	3557821 × (959-digit composite with no known proper factor, SNFS difficulty is 969.594, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1182&c0=%2B&LM=&SA=)
Φ₃₅₄₆(17)	420878287 × 5406628753 × 7195614121 × 32800804957 × (1409-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=%2B&LM=&SA=)
Φ₇₀₉₂(17)	21277 × 1560241 × 2654148561193 × (2872-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=3546&c0=%2B&LM=&SA=)

For the number 26¹⁸⁸⁶−1, it is the product of Φ_d(26) with positive integers d dividing 1886 (i.e. d = 1, 2, 23, 41, 46, 82, 943, 1886), and the factorization of Φ_d(26) for these positive integers d are:

from	currently known prime factorization
Φ₁(26)	5²
Φ₂(26)	3³
Φ₂₃(26)	13709 × 1086199 × 1528507873 × 615551139461
Φ₄₁(26)	83 × 2633923 × (49-digit prime)
Φ₄₆(26)	47 × 1157729 × 378673381 × 629584013567417
Φ₈₂(26)	9677 × 1532581 × (47-digit prime)
Φ₉₄₃(26)	384118835398327 × (1231-digit composite with no known proper factor, SNFS difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=-&LM=&SA=)
Φ₁₈₈₆(26)	(1246-digit composite with no known proper factor, SNFS difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=%2B&LM=&SA=)

For the number 36³⁹³³−1 = 6⁷⁸⁶⁶−1, it is the product of Φ_d(6) with positive integers d dividing 7866 (i.e. d = 1, 2, 3, 6, 9, 18, 19, 23, 38, 46, 57, 69, 114, 138, 171, 207, 342, 414, 437, 874, 1311, 2622, 3933, 7866), and the factorization of Φ_d(6) for these positive integers d are:

from	currently known prime factorization
Φ₁(6)	5
Φ₂(6)	7
Φ₃(6)	43
Φ₆(6)	31
Φ₉(6)	19 × 2467
Φ₁₈(6)	46441
Φ₁₉(6)	191 × 638073026189
Φ₂₃(6)	47 × 139 × 3221 × 7505944891
Φ₃₈(6)	1787 × 48713705333
Φ₄₆(6)	113958101 × 990000731
Φ₅₇(6)	47881 × 820459 × 219815829325921729
Φ₆₉(6)	11731 × 1236385853432057889667843739281
Φ₁₁₄(6)	457 × 137713 × 190324492938225748951
Φ₁₃₈(6)	24648570768391 × 816214079084081564521
Φ₁₇₁(6)	19 × 25896916098621777025320461067950269867 × (46-digit prime)
Φ₂₀₇(6)	399097 × (98-digit prime)
Φ₃₄₂(6)	62174327387790051073 × (65-digit prime)
Φ₄₁₄(6)	4811469913 × 61040960263 × 25280883279243199352415750302719 × (51-digit prime)
Φ₄₃₇(6)	989723472495640900314985156529340457 × (273-digit composite with no known proper factor, SNFS difficulty is 340.830, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=437&c0=-&LM=&SA=, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10³⁰⁰), see http://factordb.com/sequences.php?se=1&aq=%286%5E437-1%295%2F%286%5E23-1%29%2F%286%5E19-1%29&action=all&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM* effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 10³⁰)
Φ₈₇₄(6)	(309-digit prime, for its ECPP primality certificate see http://factordb.com/cert.php?id=1100000000019287760, and for its certificate chain see http://factordb.com/certchain.php?fid=1100000000019287760&action=all&fr=0&to=100)
Φ₁₃₁₁(6)	100745107 × 1719861571 × 2376829061449 × (587-digit composite with no known proper factor, SNFS difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=-&LM=&SA=)
Φ₂₆₂₂(6)	41953 × 266030354191322260711 × (592-digit composite with no known proper factor, SNFS difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=%2B&LM=&SA=)
Φ₃₉₃₃(6)	7867 × (1845-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=-&LM=&SA=)
Φ₇₈₆₆(6)	(1849-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=%2B&LM=&SA=)

For the files in this page:

File "certificate b n": The primality certificate for the nth minimal prime in base b (local copy from factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), after downloading these files, these files should be renamed to ".out" files, e.g. file "certificate9_149" is the primality certificate for the 149th minimal prime in base 9, i.e. the primality certificate for the prime 76₃₂₉2 in base 9, which equals the prime (31×9³³⁰−19)/4.

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