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

3, 7, 13, 43, 137, 63061489

Offset

1,1

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

Table of n, a(n) for n=1..6.

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

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 *)

Crossrefs

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

Sequence in context: A174241 A289556 A086208 * A020641 A358547 A062736

Adjacent sequences: A090965 A090966 A090967 * A090969 A090970 A090971

Keyword

nonn,hard,more

Author

Robert G. Wilson v, Feb 27 2004

Status

approved