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A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1. 87
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

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 3.25*10^17 as established by PrimeGrid (see link below). - Max Alekseyev, Nov 26 2014

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.


R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.

R. 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.

Y. Hellegouarch, "Invitation aux mathematiques 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.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 263.

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


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

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

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

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

D. X. Charles, On Wieferich Primes

R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.

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

John Blythe Dobson, A note on the two known Wieferich Primes

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

F. G. Dorais and D. W. Klyve, A Wieferich Prime Search up to 6.7*10^15, Journal of Integer Sequences, Vol. 14, 2011.

Will Edgington, Mersenne Page.

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

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

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)

W. Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die Reine und Angewandte Mathematik 292 (1977): 196-200.

J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp., 75 (2005), 1559-1563.

P. Lezak, Software for searching Wieferich primes - Felix Fröhlich, Jul 13 2014

C. McLeman, PlanetMath.org, Wieferich prime

Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents

A. Ostafe and I. Shparlinski, Pseudorandomness and Dynamics of Fermat Quotients, arXiv:1001.1504 [math.NT], 2010.

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

M. Rodenkirch, PRPNet (The PRPNet package includes wwww, a program that can search for Wieferich and Wall-Sun-Sun primes.) - Felix Fröhlich, Jul 13 2014

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

J. Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, arXiv 2011.

J. 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.

PrimeGrid, Wieferich Prime Search statistics

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv:0806.3412

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

Wieferich Home Page, Search for Wieferich primes

Wikipedia, Wieferich prime

P. 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: # UlrSchimke(AT)aol.com, Nov 01, 2001


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



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=10^9; 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


See A007540 for a similar problem.

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).

Cf. A001567, A077816, A001008, A039951, A049094, A126196, A126197, A178815, A178844, A178871, A178900, A246503

Sequence in context: A038469 A246503 A077816 * A203858 A115192 A091674

Adjacent sequences:  A001217 A001218 A001219 * A001221 A001222 A001223




N. J. A. Sloane



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Last modified December 19 10:51 EST 2014. Contains 252207 sequences.