A196733 Primes q = 2*p+1 for which there are primes b < c < p such that b^p == c^p == 1 (mod q^2).

555383, 1767407, 2103107, 7400567, 12836987, 14668163, 15404867, 16238303, 19572647, 25003799, 26978663, 27370727, 35182919, 36180527, 38553023, 39714083, 52503587, 53061143, 53735699, 55072427, 63302159, 70728839, 77199743, 77401679, 86334299, 97298759, 97375319

Offset: 1

Author

M. F. Hasler, Oct 05 2011

Keywords

• nonn

Comments

From D. Broadhurst, Oct 05 2011: (Start)

(p,q) is a Sophie Germain prime pair; (b,q) and (c,q) are Wieferich prime pairs; each of (b,c) is a square modulo q^2.

The sequence is now complete up to the 51st term, q=199065467.

It is a subsequence of A196511, where the latter does not require that q=2*p+1, is complete only up q=27370727, and contains q=2452757 and q=22796069, with q=4*p+1, (cf. link to post on "primenumbers" group), found by a simple analysis of Mossinghoff's results on Wieferich primes (cf. link).

With thanks to Mike Oakes. (End)

Links

• D. J. Broadhurst, Table of n, a(n) for n = 1..51
• D. J. Broadhurst et al., Re: Square factors of b^p-1 on yahoo group "primenumbers", Sept.-Oct. 2011
• David Broadhurst and others, Square factors of b^p-1, digest of 81 messages in primenumbers Yahoo group, Sep 22 - Nov 29, 2011.
• Michael Mossinghoff, Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices, as of Feb 12 2009.

Crossrefs

Cf. A005384, A005385, A124121, A124122, A196511.