A123692 -id:A123692 - cp's OEIS frontend

A305332 Multiplicative order of 5 (mod A123692(n)^2).

1, 10385, 40486, 13367790, 1645333506, 6692367336, 11796759175

Offset: 1

Felix Fröhlich, May 30 2018

From Eric Chen, Jun 07 2018: (Start)
    b    known Wieferich primes in base b (multiplicative order of b mod these primes (also these primes^2)) (if the order is p-1, then b is a primitive root to mod this prime (but not mod this prime^2), see A055578)
  1093 (364), 3511 (1755)
  11 (5), 1006003 (1006002)
  1093 (182), 3511 (1755)
  2 (1), 20771 (10385), 40487 (40486), 53471161 (13367790), 1645333507 (1645333506), 6692367337 (6692367336), 188748146801 (11796759175)
  66161 (66160), 534851 (106970), 3152573 (788143)
  5 (4), 491531 (245765)
  3 (2), 1093 (364), 3511 (585)
  2 (1), 11 (5), 1006003 (503001)
  3 (1), 487 (486), 56598313 (56598312)
  71 (70)
  2693 (2692), 123653 (123652)
  2 (1), 863 (862), 1747591 (873795)
  29 (28), 353 (352), 7596952219 (7596952218)
  29131 (29130), 119327070011 (59663535005)
  1093 (91), 3511 (1755)
  2 (1), 3 (2), 46021 (7670), 48947 (24473), 478225523351 (478225523350)
  5 (4), 7 (3), 37 (36), 331 (110), 33923 (33922), 1284043 (428014)
  3 (1), 7 (6), 13 (12), 43 (42), 137 (68), 63061489 (63061488)
  281 (140), 46457 (46456), 9377747 (9377746), 122959073 (122959072)
  2 (1)
  13 (3), 673 (224), 1595813 (797906), 492366587 (246183293), 9809862296159 (44999368331)
  13 (6), 2481757 (827252), 13703077 (13703076), 15546404183 (7773202091), 2549536629329 (2549536629328)
  5 (2), 25633 (6408)
These orders n will satisfy that Phi_n(b) is divisible by p^2, where Phi is the cyclotomic polynomial. (Usually, Phi_n(b) is squarefree, but these are all exceptions; i.e., if p^2 divides Phi_n(b) (except the case p = 2, n = 2 and b == 3 (mod 4)), then p is a Wieferich prime in base b.)
(End)

Cf. A123692, A211241, A305331, A305333.

v=[2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801]; for(k=1, #v, print1(znorder(Mod(5, v[k]^2)), ", "))

a(n) = A305331(A123692(n)).

A305333 Let p be the n-th base-5 Wieferich prime (A123692). a(n) is the value of p-1 divided by the multiplicative order of 5 (mod p^2).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 16

Offset: 1

Felix Fröhlich, May 30 2018

Cf. A123692, A211241, A305331, A305332.

v=[2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801]; for(k=1, #v, my(ord=znorder(Mod(5, v[k]^2))); print1((v[k]-1)/ord, ", "))

a(n) = (A123692(n)-1)/A305332(n).

A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

Original entry on oeis.org

1093, 3511

Offset: 1

N. J. A. Sloane

Sequence is believed to be infinite.
Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre, Jan 09 2003
Graves and Murty (2013) improved Silverman's result by showing that for any fixed k > 1, the abc-conjecture implies that there are infinitely many primes == 1 (mod k) which are not in the sequence. - Jonathan Sondow, Jan 21 2013
The squares of these numbers are Fermat pseudoprimes to base 2 (A001567) and Catalan pseudoprimes (A163209). - T. D. Noe, May 22 2003
Primes p that divide the numerator of the harmonic number H((p-1)/2); that is, p divides A001008((p-1)/2). - T. D. Noe, Mar 31 2004
In a 1977 paper, Wells Johnson, citing a suggestion from Lawrence Washington, pointed out the repetitions in the binary representations of the numbers which are one less than the two known Wieferich primes; i.e., 1092 = 10001000100 (base 2); 3510 = 110110110110 (base 2). It is perhaps worth remarking that 1092 = 444 (base 16) and 3510 = 6666 (base 8), so that these numbers are small multiples of repunits in the respective bases. Whether this is mathematically significant does not appear to be known. - John Blythe Dobson, Sep 29 2007
A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev, Jul 09 2008, Aug 24 2008
It is believed that p^2 does not divide 3^(p-1) - 1 if p = a(n). This is true for n = 1 and 2. See A178815, A178844, A178900, and Ostafe-Shparlinski (2010) Section 1.1. - Jonathan Sondow, Jun 29 2010
These primes also divide the numerator of the harmonic number H(floor((p-1)/4)). - H. Eskandari (hamid.r.eskandari(AT)gmail.com), Sep 28 2010
1093 and 3511 are prime numbers p satisfying congruence 429327^(p-1) == 1 (mod p^2). Why? - Arkadiusz Wesolowski, Apr 07 2011. Such bases are listed in A247208. - Max Alekseyev, Nov 25 2014. See A269798 for all such bases, prime and composite, that are not powers of 2. - Felix Fröhlich, Apr 07 2018
A196202(A049084(a(1))) = A196202(A049084(a(2))) = 1. - Reinhard Zumkeller, Sep 29 2011
If q is prime and q^2 divides a prime-exponent Mersenne number, then q must be a Wieferich prime. Neither of the two known Wieferich primes divide Mersenne numbers. See Will Edgington's Mersenne page in the links below. - Daran Gill, Apr 04 2013
There are no other terms below 4.97*10^17 as established by PrimeGrid (see link below). - Max Alekseyev, Nov 20 2015. The search was done via PrimeGrid's PRPNet and the results were not double-checked. Because of the unreliability of the testing, the search was suspended in May 2017 (cf. Goetz, 2017). - Felix Fröhlich, Apr 01 2018. On Nov 28 2020, PrimeGrid has resumed the search (cf. Reggie, 2020). - Felix Fröhlich, Nov 29 2020. As of Dec 29 2022, PrimeGrid has completed the search to 2^64 (about 1.8 * 10^19) and has no plans to continue further. - Charles R Greathouse IV, Sep 24 2024
Are there other primes q >= p such that q^2 divides 2^(p-1)-1, where p is a prime? - Thomas Ordowski, Nov 22 2014. Any such q must be a Wieferich prime. - Max Alekseyev, Nov 25 2014
Primes p such that p^2 divides 2^r - 1 for some r, 0 < r < p. - Thomas Ordowski, Nov 28 2014, corrected by Max Alekseyev, Nov 28 2014
For some reason, both p=a(1) and p=a(2) also have more bases b with 1 < b < p that make b^(p-1) == 1 (mod p^2) than any smaller prime p; in other words, a(1) and a(2) belong to A248865. - Jeppe Stig Nielsen, Jul 28 2015
Let r_1, r_2, r_3, ..., r_i be the set of roots of the polynomial X^((p-1)/2) - (p-3)! * X^((p-3)/2) - (p-5)! * X^((p-5)/2) - ... - 1. Then p is a Wieferich prime iff p divides sum{k=1, p}(r_k^((p-1)/2)) (see Example 2 in Jakubec, 1994). - Felix Fröhlich, May 27 2016
Arthur Wieferich showed that if p is not a term of this sequence, then the First Case of Fermat's Last Theorem has no solution in x, y and z for prime exponent p (cf. Wieferich, 1909). - Felix Fröhlich, May 27 2016
Let U_n(P, Q) be a Lucas sequence of the first kind, let e be the Legendre symbol (D/p) and let p be a prime not dividing 2QD, where D = P^2 - 4*Q. Then a prime p such that U_(p-e) == 0 (mod p^2) is called a "Lucas-Wieferich prime associated to the pair (P, Q)". Wieferich primes are those Lucas-Wieferich primes that are associated to the pair (3, 2) (cf. McIntosh, Roettger, 2007, p. 2088). - Felix Fröhlich, May 27 2016
Any repeated prime factor of a term of A000215 is a term of this sequence. Thus, if there exist infinitely many Fermat numbers that are not squarefree, then this sequence is infinite, since no two Fermat numbers share a common factor. - Felix Fröhlich, May 27 2016
If the Diophantine equation p^x - 2^y = d has more than one solution in positive integers (x, y), with (p, d) not being one of the pairs (3, 1), (3, -5), (3, -13) or (5, -3), then p is a term of this sequence (cf. Scott, Styer, 2004, Corollary to Theorem 2). - Felix Fröhlich, Jun 18 2016
Odd primes p such that Chi_(D_0)(p) != 1 and Lambda_p(Q(sqrt(D_0))) != 1, where D_0 < 0 is the fundamental discriminant of the imaginary quadratic field Q(sqrt(1-p^2)) and Chi and Lambda are Iwasawa invariants (cf. Byeon, 2006, Proposition 1 (i)). - Felix Fröhlich, Jun 25 2016
If q is an odd prime, k, p are primes with p = 2*k+1, k == 3 (mod 4), p == -1 (mod q) and p =/= -1 (mod q^3) (Jakubec, 1998, Corollary 2 gives p == -5 (mod q) and p =/= -5 (mod q^3)) with the multiplicative order of q modulo k = (k-1)/2 and q dividing the class number of the real cyclotomic field Q(Zeta_p + (Zeta_p)^(-1)), then q is a term of this sequence (cf. Jakubec, 1995, Theorem 1). - Felix Fröhlich, Jun 25 2016
From Felix Fröhlich, Aug 06 2016: (Start)
Primes p such that p-1 is in A240719.
Prime terms of A077816 (cf. Agoh, Dilcher, Skula, 1997, Corollary 5.9).
p = prime(n) is in the sequence iff T(2, n) > 1, where T = A258045.
p = prime(n) is in the sequence iff an integer k exists such that T(n, k) = 2, where T = A258787. (End)
Conjecture: an integer n > 1 such that n^2 divides 2^(n-1)-1 must be a Wieferich prime. - Thomas Ordowski, Dec 21 2016
The above conjecture is equivalent to the statement that no "Wieferich pseudoprimes" (WPSPs) exist. While base-b WPSPs are known to exist for several bases b > 1 other than 2 (see for example A244752), no base-2 WPSPs are known. Since two necessary conditions for a composite to be a base-2 WPSP are that, both, it is a base-2 Fermat pseudoprime (A001567) and all its prime factors are Wieferich primes (cf. A270833), as shown in the comments in A240719, it seems that the first base-2 WPSP, if it exists, is probably very large. This appears to be supported by the guess that the properties of a composite to be a term of A001567 and of A270833 are "independent" of each other and by the observation that the scatterplot of A256517 seems to become "less dense" at the x-axis parallel line y = 2 for increasing n. It has been suggested in the literature that there could be asymptotically about log(log(x)) Wieferich primes below some number x, which is a function that grows to infinity, but does so very slowly. Considering the above constraints, the number of WPSPs may grow even more slowly, suggesting any such number, should it exist, probably lies far beyond any bound a brute-force search could reach in the forseeable future. Therefore I guess that the conjecture may be false, but a disproof or the discovery of a counterexample are probably extraordinarily difficult problems. - Felix Fröhlich, Jan 18 2019
Named after the German mathematician Arthur Josef Alwin Wieferich (1884-1954). a(1) = 1093 was found by Waldemar Meissner in 1913. a(2) = 3511 was found by N. G. W. H. Beeger in 1922. - Amiram Eldar, Jun 05 2021
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(2^(1/k)) is Z[2^(1/k)] if and only if k does not have a prime factor in this sequence (k is in A342390). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer, but it is not in Z[2^(1/1093)];
  (1 + 2^(1755/3511) + 2^(2*1755/3511) + ... + 2^(3510*1755/3511))/3511 is an algebraic integer, but it is not in Z[2^(1/3511)]. (End)

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.
Richard K. Guy, Unsolved Problems in Number Theory, A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 91.
Yves Hellegouarch, "Invitation aux mathématiques de Fermat Wiles", Dunod, 2eme Edition, pp. 340-341.
Pace Nielsen, Wieferich primes, heuristics, computations, Abstracts Amer. Math. Soc., 33 (#1, 20912), #1077-11-48.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 263.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 230-234.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 163.

Takashi Agoh, Karl Dilcher and Ladislav Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory 66(1), 1997, 29-50.
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F. G. Dorais, WPSE - A Wieferich Prime Search Engine (A program to search Wieferich primes written by F. G. Dorais.) - _Felix Fröhlich_, Jul 13 2014
François G. Dorais and Dominic W. Klyve, A Wieferich Prime Search up to 6.7*10^15, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.2.
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Stanislav Jakubec, Connection between the Wieferich congruence and divisibility of h+, Acta Arithmetica, Vol. 71, No. 1 (1995), pp. 55-64.
Stanislav Jakubec, On divisibility of the class number h+ of the real cyclotomic fields of prime degree l, Mathematics of Computation, Vol. 67, No. 221 (1998), pp. 369-398.
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Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
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Cf. similar primes related to the first case of Fermat's last theorem: A007540, A088164.
Sequences "primes p such that p^2 divides X^(p-1)-1": A014127 (X=3), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).
Cf. A001567, A002323, A077816, A001008, A039951, A049094, A126196, A126197, A178815, A178844, A178871, A178900, A246503, A247208, A269798.

Filtered([1..50000],p->IsPrime(p) and (2^(p-1)-1) mod p^2 =0); # Muniru A Asiru, Apr 03 2018

import Data.List (elemIndices)
a001220 n = a001220_list !! (n-1)
a001220_list = map (a000040 . (+ 1)) $ elemIndices 1 a196202_list
-- Reinhard Zumkeller, Sep 29 2011

[p : p in PrimesUpTo(310000) | IsZero((2^(p-1) - 1) mod (p^2))]; // Vincenzo Librandi, Jan 19 2019

wieferich := proc (n) local nsq, remain, bin, char: if (not isprime(n)) then RETURN("not prime") fi: nsq := n^2: remain := 2: bin := convert(convert(n-1, binary),string): remain := (remain * 2) mod nsq: bin := substring(bin,2..length(bin)): while (length(bin) > 1) do: char := substring(bin,1..1): if char = "1" then remain := (remain * 2) mod nsq fi: remain := (remain^2) mod nsq: bin := substring(bin,2..length(bin)): od: if (bin = "1") then remain := (remain * 2) mod nsq fi: if remain = 1 then RETURN ("Wieferich prime") fi: RETURN ("non-Wieferich prime"): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 01 2001

Select[Prime[Range[50000]],Divisible[2^(#-1)-1,#^2]&]  (* Harvey P. Dale, Apr 23 2011 *)
Select[Prime[Range[50000]],PowerMod[2,#-1,#^2]==1&] (* Harvey P. Dale, May 25 2016 *)

N=10^4; default(primelimit,N);
forprime(n=2,N,if(Mod(2,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

from sympy import prime
from gmpy2 import powmod
A001220_list = [p for p in (prime(n) for n in range(1,10**7)) if powmod(2,p-1,p*p) == 1]
# Chai Wah Wu, Dec 03 2014

(A178815(A000720(p))^(p-1) - 1) mod p^2 = A178900(n), where p = a(n). - Jonathan Sondow, Jun 29 2010

Odd primes p such that A002326((p^2-1)/2) = A002326((p-1)/2). See A182297. - Thomas Ordowski, Feb 04 2014

A014127 Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1.

Original entry on oeis.org

11, 1006003

Offset: 1

N. J. A. Sloane

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
These primes are so named after the celebrated result of Mirimanoff in 1910 (see below) that for a failure of the first case of Fermat's Last Theorem, the exponent p must satisfy the criterion stated in the definition. Lerch (see below) showed that these primes also divide the numerator of the harmonic number H(floor(p/3)). This is analogous to the fact that Wieferich primes (A001220) divide the numerator of the harmonic number H((p-1)/2). - John Blythe Dobson, Mar 02 2014, Apr 09 2015
The prime 1006003 was apparently discovered by K. E. Kloss (cf. Kloss, 1965) according to various sources. - Felix Fröhlich, Dec 08 2020
If there is no term other than 11 and 1006003, then the only solution (a, w, x, y, z) to the diophantine equation a^w + a^x = 3^y + 3^z is (5, 1, 1, 2, 3) (cf. Scott, Styer, 2006, Lemma 12). - Felix Fröhlich, Dec 10 2020
Named after the Russian mathematician Dmitry Semionovitch Mirimanoff (1861-1945). - Amiram Eldar, Jun 10 2021

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152-153.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.
Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Chris K. Caldwell, Fermat Quotient, The Prime Glossary.
John Blythe Dobson, On the special harmonic numbers H_floor(p/9) and H_floor(p/18) modulo p, arXiv:2302.02027 [math.NT], 2023.
François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq., Vol. 14 (2011), Article 11.9.2, 1-14.
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp., Vol. 74, No. 250 (2005), pp. 927-936.
K. E. Kloss, Some Number-Theoretic Calculations, Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (Oct.-Dec. 1965), pp. 335-336.
Mathias Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1) - 1)/p == q(a), Mathematische Annalen, Vol. 60 (1905), pp. 471-490.
D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris, Vol. 150 (1910), pp. 204-206. Revised as Sur le dernier théorème de Fermat, Journal für die reine und angewandte Mathematik, Vol. 139 (1911), pp. 309-324.
Planet Math, Wieferich Primes.
Reese Scott and Robert Styer, On the generalized Pillai equation +-a^x +-b^y = c, Journal of Number Theory, Vol. 118, No. 2 (2006), pp. 236-265.

Cf. A039951, A096082.

Sequences "primes p such that p^2 divides X^(p-1)-1": A001220 (X=2), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).

Select[Prime[Range[1000000]], PowerMod[3, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

N=10^9; default(primelimit,N);
forprime(n=2,N,if(Mod(3,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

from sympy import prime
from gmpy2 import powmod
A014127_list = [p for p in (prime(n) for n in range(1,10**7)) if powmod(3,p-1,p*p) == 1] # Chai Wah Wu, Dec 03 2014

Edited by Max Alekseyev, Oct 20 2010

Updated by Max Alekseyev, Jan 29 2012

A039951 a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1.

Original entry on oeis.org

2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3

Offset: 1

David W. Wilson

a(n^k) <= a(n) for any n,k > 1.
a(n) is currently unknown for n in {47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, ...}. - Richard Fischer, Jul 15 2021
a(47) > 1.4*10^14, a(72) > 1.4*10^14 (see Fischer's tables).
For all nonnegative integers n and k, a(n^(n^k)) = a(n) (see Puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014

C. K. Caldwell, The Prime Glossary, Fermat quotient.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
Richard Fischer, Update Table of n, July 15 2021.
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.
Carlos Rivera, Puzzle 762. Conjecture from Ribenboim's book, The Prime Puzzles and Problems Connection.
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (with missing terms)

Cf. A001220, A045616, A096082, A014127, A123692, A123693, A174422.

Table[p = 2; While[! Divisible[n^(p - 1) - 1, p^2], p = NextPrime@ p]; p, {n, 33}] (* Michael De Vlieger, Nov 24 2016 *)
f[n_] := Block[{p = 2}, While[ PowerMod[n, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 33] (* Robert G. Wilson v, Jul 18 2018 *)

a(n)={forprime(p=2, oo, if(Mod(n, p^2)^(p-1)==1, return(p))); oo} \\ Felix Fröhlich, Jul 24 2014

a(4k+1) = 2.

a(n) = A096082(n) for all n > 1 that are not of the form 4k+1. Note that A096082 begins with n = 2. [Corrected and clarified by Jonathan Sondow, Jun 17-18 2010]

a(34)-a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004
Entry revised by N. J. A. Sloane, Nov 30 2006
Edited by Max Alekseyev, Oct 06, Oct 09 2009
Edited and updated by Max Alekseyev, Jan 29 2012

A123693 Primes p such that p^2 divides 7^(p-1) - 1.

Original entry on oeis.org

5, 491531

Offset: 1

Max Alekseyev, Oct 07 2006

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
F. G. Dorais and D. Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.

Cf. A001220, A014127, A123692, A123693, A128667, A128668, A090968, A039951, A128669

Select[Prime[Range[1000000]], PowerMod[7, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

Updated by Max Alekseyev, Jan 29 2012

A128667 Primes p such that p^2 divides 13^(p-1) - 1.

Original entry on oeis.org

2, 863, 1747591

Offset: 1

Alexander Adamchuk, Mar 26 2007

No further terms up to 3.127*10^13.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
C. K. Caldwell, Fermat quotient
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)

Cf. A001220, A014127, A123692, A123693, A128668, A090968, A128669, A039951

Select[Prime[Range[5*10^7]], Mod[ 13^(# - 1) - 1, #^2] == 0 &] (* G. C. Greubel, Jan 18 2018 *)

A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

Original entry on oeis.org

3, 487, 56598313

Offset: 1

Helmut Richter, Dec 11 1999

Primes p such that the decimal fraction 1/p has same period length as 1/p^2, i.e., the multiplicative order of 10 modulo p is the same as the multiplicative order of 10 modulo p^2. [extended by Felix Fröhlich, Feb 05 2017]
No further terms below 1.172*10^14 (as of Feb 2020, cf. Fischer's table).
56598313 was announced in the paper by Brillhart et al. - Helmut Richter, May 17 2004
A265012(A049084(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2015
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(10^(1/k)) is Z[10^(1/k)] if and only if k does not have a prime factor in this sequence. See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 10^(1/3) + 10^(2/3))/3 is an algebraic integer, but it is not in Z[10^(1/3)];
  (1 + 10^(486/487) + 10^(2*486/487) + ... + 10^(486*486/487))/487 is an algebraic integer, but it is not in Z[10^(1/487)];
  (1 + 10^(56598312/56598313) + 10^(2*56598312/56598313) + ... + 10^(56598312*56598312/56598313))/56598313 is an algebraic integer, but it is not in Z[10^(1/56598313)]. (End)

J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, pp. 213-222 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, A3.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Keith Conrad, The ring of integers in a radical extension.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2).
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
Peter L. Montgomery, New solutions of a^(p-1) == 1 (mod p^2), Math. Comp. 61 (1993), 361-363.
Math Overflow, Is the smallest primitive root modulo p a primitive root modulo p^2?, Jun 09 2010.
Helmut Richter, The period length of the decimal expansion of a fraction.
Helmut Richter, The Prime Factors Of 10^486-1.
Siqiong Yao and Akira Toyohara, The length of the repeating decimal, arXiv:2507.01295 [math.NT], 2025. See p. 19.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A001220, A014127, A123692, A212583, A123693, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A039951.

Cf. A265012, A049084, A000040.

import Math.NumberTheory.Moduli (powerMod)
a045616 n = a045616_list !! (n-1)
a045616_list = filter
               (\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'
-- Reinhard Zumkeller, Nov 30 2015

A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* JungHwan Min, Feb 04 2017 *)
Select[Prime[Range[34*10^5]],PowerMod[10,#-1,#^2]==1&] (* Harvey P. Dale, Apr 10 2018 *)

lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015

A128668 Primes p such that p^2 divides 17^(p-1) - 1.

Original entry on oeis.org

2, 3, 46021, 48947, 478225523351

Offset: 1

Alexander Adamchuk, Mar 26 2007

Mossinghoff showed that there are no further terms up to 10^14.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Cryptogr. 53 (2009), 149-163.

Cf. A001220, A014127, A123692, A123693, A128667, A090968, A128669, A039951.

Select[Prime[Range[5*10^6]], Mod[ 17^(# - 1) - 1, #^2] == 0 &] (* G. C. Greubel, Jan 18 2018 *)

The prime 478225523351 was found by Richard Fischer on Oct 25 2005

Extension corrected by Jonathan Sondow, Jun 24 2010

A090968 Primes p such that p^2 divides 19^(p-1) - 1.

Original entry on oeis.org

3, 7, 13, 43, 137, 63061489

Offset: 1

Robert G. Wilson v, Feb 27 2004

Primes p such that p divides the Fermat quotient of p (with base 19). The Fermat quotient of p with base a denotes the integer q_p(a) = ( a^(p-1) - 1) / p, where p is a prime which does not divide the integer a. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 20 2005

No further terms up to 3.127*10^13.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 43, p. 17, Ellipses, Paris 2008.
Paulo Ribenboim, The Little Book Of Big Primes, Springer-Verlag, NY 1991, page 170.
Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 171. [Harvey P. Dale, Oct 17 2011]

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
C. Caldwell, Fermat quotient
W. Keller and J. Richstein Fermat quotients q_p(a) that are divisible by p

Cf. A001220, A014127, A123692, A123693, A128667, A128668, A128669, A039951, A096082

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ p = NextPrim[p]; If[PowerMod[19, p - 1, p^2] == 1, Print[p]], {n, 1, 2*10^8}]
Select[Prime[Range[4*10^6]],PowerMod[19,#-1,#^2]==1&] (* Harvey P. Dale, Nov 08 2017 *)

cp's OEIS Frontend

A305332 Multiplicative order of 5 (mod A123692(n)^2).

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A305333 Let p be the n-th base-5 Wieferich prime (A123692). a(n) is the value of p-1 divided by the multiplicative order of 5 (mod p^2).

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A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

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A014127 Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1.

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A039951 a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1.

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A123693 Primes p such that p^2 divides 7^(p-1) - 1.

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A128667 Primes p such that p^2 divides 13^(p-1) - 1.

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