The Impact of Number Theory and Computer-Aided Mathematics on Solving the Hadamard Matrix Conjecture
The Hadamard conjecture has been studied since the pioneering paper of Sylvester, “Thoughts on inverse orthogonal matrices, simultaneous sign successions, tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile work and the theory of numbers,” Phil Mag, 34 (1867), 461–475, first appeared. We review the importance of primes on those occasions that the conjecture, that for every odd number, t there exists an Hadamard matrix of order 4t – is confirmed. Although substantial advances have been made into the question of the density of this odd number, t it has still not been shown to have positive density. We survey the results of some computer-aided construction algorithms for Hadamard matrices.
Key wordsHadamard matrix Skew-Hadamard matrix Symmetric Hadamard matrix Asymptotic Hadamard matrices Prime number theorem Extended Riemann hypothesis Computer-aided construction
The author sincerely thanks Professor Igor Shparlinski and Dr Daniel Gordon for alerting her to her confusion in Sect. 3.2 and pointing out pertinent references.
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