Wieferich pairs and Barker sequences

Abstract

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001034 441 522 766 781 604, or n > 2ċ1030. This improves the lower bound on the length of a long Barker sequence by a factor of more than 107. We also show that all but fewer than 1600 integers n ≤ 4 ċ 1026 can be eliminated as the order of a circulant Hadamard matrix. These results are obtained by completing extensive searches for Wieferich prime pairs (q, p), which are defined by the relation qP-1≡ 1 mod p2 and analyzing their results in combination with a number of arithmetic restrictions on n.