A124122 Least prime p such that (p,q) is a Double Wieferich prime pair for q=A124121(n).

1093, 1006003, 1645333507, 4871, 318917, 18787

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 (least) value of p corresponding to the q's listed in increasing order (and without multiplicity) in A124121.

This is just the list of known pairs: there may be gaps.

Currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). - Alexander Adamchuk, Mar 10 2007

Links

• Y. F. Bilu, Catalan's Conjecture, Séminaire Bourbaki, 45 (2002-2003), pp. 1-26.
• Michael Mossinghoff, Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices, as of Feb 12 2009.
• Wikipedia, Wieferich pair

Crossrefs

See A124121 for values of q.

Cf. A196511, A196733.

Programs

PARI

/* The following (highly unoptimized) code misses the value a(3) but prints all other values in less than 30 seconds. */
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( p, ", ") || break))   \\ M. F. Hasler, Oct 08 2011

PARI

a(n) = my(q=prime(n), p=2); while(Mod(p, q^2)^(q-1)!=1 || Mod(q, p^2)^(p-1)!=1, p=nextprime(p+1)); p \\ Felix Fröhlich, Jan 04 2016