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 . 10(30). This improves the lower bound on the length of a long Barker sequence by a factor of more than 10(7). We also show that all but fewer than 1600 integers n <= 4 . 10(26) 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 q(p-1) equivalent to 1 mod p(2), and analyzing their results in combination with a number of arithmetic restrictions on n.
barker sequence circulant hadamard matrix wieferich prime pair