A124121 Primes q which are the lesser of a double Wieferich prime pair. (List of known values of q in increasing order, without multiplicity.)

2, 3, 5, 83, 911, 2903

Offset: 1

Author

N. J. A. Sloane, following an email from Robert G. Wilson v, Nov 30 2006

Keywords

• nonn
• hard
• more

Comments

Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2). This sequence gives the primes q which are the lesser member (listed second) of such pairs, in increasing order, but without multiplicity:

For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). In this sequence, 5 is only listed once, as a(3), and only the lesser value p=1645333507 is listed as A124122(3).

This is just the list of known pairs: there may be gaps, i.e. missing primes.

Links

• Yuri F. Bilu, Catalan's Conjecture, Seminaire Bourbaki, (2002-2003).
• Michael Mossinghoff, Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices, as of Feb 12 2009.
• Eric Weisstein's World of Mathematics, Double Wieferich Prime Pair
• Wikipedia, Wieferich pair

Crossrefs

See A124122 for values of p.

Cf. A196511, A196733.

Programs

PARI

/* The following (highly unoptimized) code misses the value q=5 (corresponding to a very large value of p) */
default(primelimit,1010000); forprime(q=1, default(primelimit), forprime(p=q+1, default(primelimit),  Mod(p, q^2)^(q-1)-1 & next; Mod(q, p^2)^(p-1)-1 || print1( q", ") || break))  \\ M. F. Hasler, Oct 08 2011