1,1

Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 09 2003

The squares of these numbers are Fermat pseudoprimes to base 2 (A001567). - T. D. Noe (noe(AT)sspectra.com), 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 (noe(AT)sspectra.com), 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 (j.dobson(AT)uwinnipeg.ca), Sep 29 2007

A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 09 2008, Aug 24 2008

Dorais and Klyve (see reference) reported on November 27, 2008, that there are no other Wieferich primes up to 6.7*10^15. [From Peter Luschny (peter(AT)luschny.de), Feb 10 2009]

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.

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

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

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

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

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv.org/abs/0806.3412

Joerg Arndt, Fxtbook

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

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

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.

C. McLeman, PlanetMath.org, Wieferich prime

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

F.G. Dorais and D.W. Klyve, Near Wieferich Primes up to 6.7*10^15, November 27, 2008, PDF [From Peter Luschny (peter(AT)luschny.de), Feb 10 2009]

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

Select[Prime[Range[10^3*5]], Round[(2^(#-1)-1)/#^2]==((2^(#-1)-1)/#^2) &] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), May 01 2008)

See A007540 for a similar problem. Cf. A001567, A077816.

Sequence in context: A023698 A038469 A077816 this_sequence A115192 A091674 A022197

Adjacent sequences: A001217 A001218 A001219 this_sequence A001221 A001222 A001223

nonn,hard,bref,nice,more

N. J. A. Sloane (njas(AT)research.att.com).

Sequence is believed to be infinite, although there are no other terms < 4*10^12.

Wieferich Home Page link from Filip Zaludek (filip.zaludek(AT)gtsnovera.cz), Feb 05 2008