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The impact of number theory and computer aided mathematics on solving the Hadamard matrix conjecture

Chapter


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.

Publication Date


  • 2013

Citation


  • Seberry, J. (2013). The impact of number theory and computer aided mathematics on solving the Hadamard matrix conjecture. In J. M. Borwein, I. Shparlinski & W. Zudilin (Eds.), Number Theory and Related Fields (pp. 313-320). New York, United States: Springer.

International Standard Book Number (isbn) 13


  • 9781461466413

Scopus Eid


  • 2-s2.0-84883379485

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1507

Book Title


  • Number Theory and Related Fields

Start Page


  • 313

End Page


  • 320

Place Of Publication


  • New York, United States

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.

Publication Date


  • 2013

Citation


  • Seberry, J. (2013). The impact of number theory and computer aided mathematics on solving the Hadamard matrix conjecture. In J. M. Borwein, I. Shparlinski & W. Zudilin (Eds.), Number Theory and Related Fields (pp. 313-320). New York, United States: Springer.

International Standard Book Number (isbn) 13


  • 9781461466413

Scopus Eid


  • 2-s2.0-84883379485

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1507

Book Title


  • Number Theory and Related Fields

Start Page


  • 313

End Page


  • 320

Place Of Publication


  • New York, United States