LMS Journal of Computation and Mathematics

Wieferich pairs and Barker sequences, II

Peter Borweina1 and Michael J. Mossinghoffa2

a1 Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada email pborwein@sfu.ca

a2 Department of Mathematics, Davidson College, Davidson, NC 28035-6996, USA email mimossinghoff@davidson.edu

Abstract

We show that if a Barker sequence of length exists, then either n 3 979 201 339 721 749 133 016 171 583 224 100, or . This improves the lower bound on the length of a long Barker sequence by a factor of nearly . We also obtain eighteen additional integers that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates . These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on , to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.

(Received July 02 2013)

(Online publication April 2014)

2010 Mathematics Subject Classification

  • 05B20;
  • 94A55 (primary);
  • 05-04;
  • 05B10;
  • 11A07 (secondary)