A212583 Primes p such that p^2 divides 6^(p-1) - 1.

66161, 534851, 3152573

Offset: 1

Author

Felix Fröhlich, May 22 2012

Keywords

• nonn
• hard
• bref
• more

Comments

Base 6 Wieferich primes.

Next term > 4.119*10^13. [See Fischer link]

References

• P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, page 347

Links

• Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table p. 5.
• François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
• Richard Fischer, Thema: Fermatquotient B^P-1 == 1 mod (P^2)
• Wilfrid Keller and Jörg Richstein, Fermat quotients q_p(a) that are divisible by p.
• Eric Weisstein, Fermat Quotient, MathWorld
• Wikipedia, Base-a Wieferich primes

Crossrefs

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

Programs

Mathematica

Select[Prime[Range[1000000]], PowerMod[6, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

PARI

N=10^9; default(primelimit,N);
forprime(n=2,N,if(Mod(6,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013