# Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices

## Michael Mossinghoff

This page contains data associated with the article Wieferich pairs and Barker sequences, II, by P. Borwein and M. J. Mossinghoff, including lists of Wieferich prime pairs associated with computations performed for this article, and integers n that have not been eliminated as the possible length of a long Barker sequence, or as the order of a large circulant Hadamard matrix.

Recall that a Wieferich prime pair (q, p) has the property that qp−1 = 1 mod p2. A Barker sequence is a finite sequence {ai}, each term ±1, for which each sum of the form ∑i aiai+k with k ≠ 0 is −1, 0, or 1. 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.

1. The prior version of this site, containing the data associated with the earlier article Wieferich primes and Barker sequences (M. J. Mossinghoff, Des. Codes Cryptogr. 53 (2009), no. 3, 149-163).

2. 156927 Wieferich prime pairs (q, p) with q < p required for the construction of the directed graph D(1016.5) in the new article, but not appearing in the data listed with the earlier article. (Compressed file.)

3. The 4656 cycles appearing in the directed graph D(1016.5), described in the latest article.

4. Permissible Lengths of Barker Sequences.

5. Permissible Orders of Circulant Hadamard Matrices.
• The 1371 values of u ≤ 1013 for which n = 4m2 has not been eliminated as the possible order of a circulant Hadamard matrix.
• The 22 values of u ≤ 1013 for which n = 4m2 was eliminated as the possible order of a circulant Hadamard matrix by using Turyn's self-conjugacy test, but was not excluded by the computations performed in the prior paper, nor by the new methods of the 2012 article by K. H. Leung and B. Schmidt.