Wieferich Prime Pairs, Barker Sequences,
and Circulant Hadamard Matrices

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 q^p−1 = 1 mod p². A Barker sequence is a finite sequence {a_i}, each term ±1, for which each sum of the form ∑_i a_ia_i+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(10^16.5) 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(10^16.5), described in the latest article.


4. Permissible Lengths of Barker Sequences.
   • Only one value of n in the interval (13, 4 · 10³³] has not been eliminated as the possible length of a Barker sequence: n = 3979201339721749133016171583224100. Here, n = 4u² with u = 31540455528264605 = 5 · 13 · 29 · 41 · 2953 · 138200401.
   • Nineteen integers u < 5 · 10²⁴ which pass all known requirements for n = 4u² to be the length of a Barker sequence.
   • 541 integers u < 5 · 10⁴⁹ with Ω(u) ≤ 6 which pass all known requirements for n = 4u² to be the length of a Barker sequence.
     • Ω(u) = 4: 3 values.
     • Ω(u) = 5: 65 values.
     • Ω(u) = 6: 473 values.
   • 237807 integers u < 5 · 10⁴⁹ which pass most of the requirements for n = 4u² to be the length of a Barker sequence, but for which some computationally expensive tests have not been performed. (Compressed file.)


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.

Related Links

• Data on Circulant Hadamard Matrices, which contains data associated with the article by Bernhard Schmidt and Ka Hin 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 Wilfrid Keller and Jörg Richstein.
• Fermat quotients, by Richard Fischer.

Last modified May 31, 2013.