A123692 Primes p such that p^2 divides 5^(p-1) - 1.

2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

Offset

1,1

Comments

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.

a(6) and a(7) were found by Keller and Richstein (cf. Keller, Richstein, 2005). - Felix Fröhlich, Jan 06 2017

Prime terms of A242959. - Felix Fröhlich, Jan 06 2017

Links

Table of n, a(n) for n=1..7.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.

Chris K. Caldwell, The Prime Glossary, Fermat quotient.

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.

W. Keller and J. Richstein, Solutions of the congruence a^p-1 == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.

A. Paszkiewicz, A new prime p for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.

Mathematica

Select[Prime[Range[2500]], Divisible[5^(# - 1) - 1, #^2] &] (* Alonso del Arte, Aug 01 2014 *)
Select[Prime[Range[55*10^6]], PowerMod[5, #-1, #^2]==1&] (* The program generates the first 4 terms of the sequence. *) (* Harvey P. Dale, Jan 29 2023 *)

Prog

(PARI)
N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(5, n^2)^(n-1)==1, print1(n, ", ")));
\\ Joerg Arndt, May 01 2013

Crossrefs

Cf. A001220, A014127, A123693, A128667, A128668, A090968, A128669, A096082, A242959.

Sequence in context: A346988 A339476 A242959 * A290741 A261362 A132942

Adjacent sequences: A123689 A123690 A123691 * A123693 A123694 A123695

Keyword

hard,nonn,more

Author

Max Alekseyev, Oct 07 2006

Extensions

More terms from Alexander Adamchuk, Nov 27 2006

Updated by Max Alekseyev, Jan 29 2012

Status

approved