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The Impact of Number Theory and Computer-Aided Mathematics on Solving the Hadamard Matrix Conjecture

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Book cover Number Theory and Related Fields

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 43))

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Centre for Computer and Information Security Research, SCSSE, University of Wollongong, Wollongong, NSW, 2522, Australia

    Jennifer Seberry

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  1. Jennifer Seberry
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Correspondence to Jennifer Seberry .

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Editors and Affiliations

  1. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Jonathan M. Borwein

  2. , Department of Computing, Macquarie University, Herring Road, North Ryde, 2109, New South Wales, Australia

    Igor Shparlinski

  3. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Wadim Zudilin

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In memory of Alf van der Poorten

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Seberry, J. (2013). The Impact of Number Theory and Computer-Aided Mathematics on Solving the Hadamard Matrix Conjecture. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_16

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