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

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

Offset: 1

Author

Max Alekseyev, Oct 07 2006

Keywords

• hard
• nonn
• more

Comments

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

From Felix Fröhlich, Jan 06 2017: (Start)

a(6) and a(7) were found by Keller and Richstein (cf. Keller, Richstein, 2005).

Prime terms of A242959. (End)

From Jianing Song, Jun 21 2025: (Start)

The ring of integers of Q(5^(1/k)) is Z[5^(1/k)] if and only if k does not have a prime factor in this sequence (k is even or in A342391). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:

(1 + sqrt(5))/2 is an algebraic integer, but it is not in Z[sqrt(5)];

(1 + 5^(10385/20771) + 5^(2*10385/20771) + ... + 5^(10384*10385/20771))/20771 is an algebraic integer, but it is not in Z[5^(1/20771)];

(1 + 5^(40486/40487) + 5^(2*40486/40487) + ... + 5^(40486*40486/40487))/40487 is an algebraic integer, but it is not in Z[5^(1/40487)]. (End)

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.

Links

• 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.
• Keith Conrad, The ring of integers in a radical extension.
• 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.

Crossrefs

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

Programs

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 *)

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

Extensions

More terms from Alexander Adamchuk, Nov 27 2006

Updated by Max Alekseyev, Jan 29 2012