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A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1. 163
1093, 3511 (list; graph; refs; listen; history; text; internal format)



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

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


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.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 163.


Table of n, a(n) for n=1..2.

Takashi Agoh, Karl Dilcher and Ladislav Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory 66(1), 1997, 29-50.

Joerg Arndt, Matters Computational (The Fxtbook), p. 780.

Alex Samuel Bamunoba, A note on Carlitz Wieferich primes, Journal of Number Theory, Vol. 174 (2017), pp. 343-357;

N. G. W. H. Beeger, On a New Case of the Congruence 2^(p-1) == 1 (mod p^2), Messenger of Mathematics, Vol 51 (1922), pp. 149-150.

Dongho Byeon, Class numbers, Iwasawa invariants and modular forms, Trends in Mathematics, Vol. 9, No. 1, (2006), pp. 25-29.

Chris K. Caldwell, The Prime Glossary, Wieferich prime.

Chris K. Caldwell, Prime-square Mersenne divisors are Wieferich.

Denis Xavier Charles, On Wieferich Primes.

Richard Crandall, Karl Dilcher and Carl Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 433-449; alternative link.

Joe K. Crump, Joe's Number Theory Web, Weiferich Primes. (sic)

John Blythe Dobson, A note on the two known Wieferich Primes, 2007-2015.

John Blythe Dobson A Characterization of Wilson-Lerch Primes, Integers, Vol. 16 (2016), A51.

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.

Bruno Dular, Cycles of Sums of Integers, arXiv:1905.01765 [math.NT], 2019.

Will Edgington, Mersenne Page [from Internet Archive Wayback Machine].

M. Goetz, WSS and WFS are suspended, PrimeGrid forum, Message 107809, May 11, 2017.

Andrew Granville and K. Soundararajan, A binary additive problem of Erdos and the order of 2 mod p^2, Raman. J., Vol. 2 (1998) pp. 283-298

Hester Graves and M. Ram Murty, The abc conjecture and non-Wieferich primes in arithmetic progressions, Journal of Number Theory, Vol. 133 (2013), pp. 1809-1813.

René Gy, Extended Congruences for Harmonic Numbers, arXiv:1902.05258 [math.NT], 2019.

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)

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.

Stanislav Jakubec, The Congruence for Gauss Period, Journal of Number Theory, Vol. 48, No. 1 (1994), pp. 36-45.

Wells Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die reine und angewandte Mathematik, Vol. 292, (1977), pp. 196-200.

Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.

Joshua Knauer and Jörg Richstein, The continuing search for Wieferich primes, Math. Comp., Vol. 74, No. 251 (2005), pp. 1559-1563.

D. H. Lehmer, On Fermat's quotient, base two, Math. Comp., Vol. 36, No. 153 (1981), pp. 289-290.

Richard J. McIntosh and Eric L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., Vol 76, No. 260 (2007), pp. 2087-2094.

C. McLeman, PlanetMath.org, Wieferich prime.

Waldemar Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, Vol. 35 (1913), pp. 663-667. [Annotated scanned copy]

Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents, IACR, Report 2012/033, 2012.

Mishima Miwako and Koji Momihara, A new series of optimal tight conflict-avoiding codes of weight 3, Discrete Mathematics, Vol. 340, No. 4 (2017), pp. 617-629. See page 618.

Alina Ostafe and Igor E. Shparlinski, Pseudorandomness and Dynamics of Fermat Quotients, arXiv:1001.1504 [math.NT], 2010.

Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.

Reggie, Welcome to the Wieferich and Wall-Sun-Sun Prime Search, PrimeGrid forum, 2020.

Reese Scott and Robert Styer, On p^x - q^y = c and related three term exponential Diophantine equations with prime bases, Journal of Number Theory, Vol. 105, No. 2 (2004), pp. 212-234.

Vladimir Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008.

Joseph Silverman, Wieferich's Criterion and the abc Conjecture, J. Number Th. 30 (1988) 226-237.

Jonathan Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, arXiv:1110.3113 [math.NT], 2012.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101 (2014), pp. 243-255.

Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

Eric Weisstein's World of Mathematics, Wieferich Prime.

Eric Weisstein's World of Mathematics, abc Conjecture.

Eric Weisstein's World of Mathematics, Integer Sequence Primes.

A. Wieferich, Zum letzten Fermat'schen Theorem, Journal für die reine und angewandte Mathematik, Vol. 136 (1909), pp. 293-302.

Wikipedia, Wieferich prime.

Paul Zimmermann, Records for Prime Numbers.


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


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 *)



import Data.List (elemIndices)

a001220 n = a001220_list !! (n-1)

a001220_list = map (a000040 . (+ 1)) $ elemIndices 1 a196202_list

-- Reinhard Zumkeller, Sep 29 2011


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

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

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


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.

Sequence in context: A246503 A077816 A291961 * A265630 A355545 A291194

Adjacent sequences: A001217 A001218 A001219 * A001221 A001222 A001223




N. J. A. Sloane



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