This page contains data associated with two articles.

- Double Wieferich pairs and circulant Hadamard matrices, by B. Logan and M. J. Mossinghoff.
- Wieferich pairs and Barker sequences, II, by P. Borwein and M. J. Mossinghoff, LMS J. Comput. Math. 17 (2014), no. 1, 24-32.

Recall that a *Wieferich prime pair* (*q*, *p*) has the
property that *q*^{p−1} = 1 mod
*p*^{2}.
A *circulant Hadamard matrix* of order *n* is an *n*
× *n* matrix of ±1's whose rows are mutually orthogonal,
and each of whose rows after the first is obtained from the prior one by
cyclically shifting its elements by one position to the right.
A *Barker sequence* is a finite sequence {*a _{i}*},
each term ±1, for which each sum of the form
∑

- The 4489 values of
*u*≤ 10^{15}for which*n*= 4*u*^{2}cannot presently be eliminated as a permissible order of a circulant Hadamard matrix, from the first article cited above. - The 4501 Wieferich prime
pairs (
*q*,*p*) with*q*<*p*required for the construction of the directed graph in the first article above.

- The 8125 integers
*u*≤ 5 · 10^{49}for which 4*u*^{2}has not been eliminated as a possible order of a Barker sequence. These integers*u*pass all or most all of the known requirements on the length of such a sequence. (Some computationally expensive tests for values*u*with at least 9 prime factors have not been performed.) - 156927 Wieferich prime
pairs (
*q*,*p*) with*q*<*p*required for the construction of the directed graph in the 2014 LMS JCM article on Barker sequences, but not appearing in the data listed with the earlier article. (Compressed file.) - The 4656 cycles appearing in the directed graph constructed in the 2014 LMS JCM article on Barker sequences.

- Data on
Circulant Hadamard Matrices, which contains data associated with the
article by B. Schmidt and K. H. Leung, New restrictions on
possible orders of circulant Hadamard matrices (Des. Codes
Cryptogr.
**64**(2012), no. 1-2, 143-151). - Fermat
quotients
*q*(_{p}*a*) that are divisible by*p*, by W. Keller and J. Richstein. - Fermat quotients, by R. Fischer.
- The prior version of this site, which includes
a link to the data associated with the earlier article
Wieferich
primes and Barker sequences (M. J. Mossinghoff, Des. Codes
Cryptogr.
**53**(2009), no. 3, 149-163).

Michael Mossinghoff

mimossinghoff at davidson dot edu

Last modified August 2, 2015.