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

3, 7, 13, 43, 137, 63061489

Offset: 1

Author

Robert G. Wilson v, Feb 27 2004

Keywords

• nonn
• hard
• more

Comments

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.

References

• 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]

Links

• 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

Crossrefs

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

Programs

Mathematica

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 *)