Wieferich pairs and Barker sequences

We show that if a Barker sequence of length n > 13 exists, then either n = 189 260 468 001 034 441 522 766 781 604, or n > 2 · 1030. 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 ·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.