OFFSET
1,1
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.
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
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