Analyses of row expansions within the octic “chess” modular ring, Z8

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 5, 1999, Number 3, Pages 102—114
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Authors and affiliations

J. V. Leyendekkers
The University of Sydney, 2006, Australia

A. G. Shannon
University of Technology, Sydney, 2007, Australia

Abstract

A modular ring ℤ8 is described as an expanded form of ℤ4 in which the parities of the rows within the four classes of ℤ4 are distinguished. This octic or ‘chess’ ring is used to analyse the structure of rows characterising powers xn within the eight classes of ℤ8. With n even, three classes contain the squares, x2, but only two contain xn when n > 2. With n odd, even xn are confined to 08, but odd numbers, xn, have the same class as x. The rows containing squares are related to first order recurrences, while rows containing xn, n > 2, are related to third and higher order recurrences. Because of this it is conjectured that the sum or difference of xn, yn cannot fit into a row which is characteristic of a power and hence cannot equal a power when n > 2.

AMS Classification

  • 11B37
  • 11A07
  • 06F25

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Cite this paper

Leyendekkers, J. V. & Shannon, A. G. (1999). Analyses of row expansions within the octic “chess” modular ring, Z8. Notes on Number Theory and Discrete Mathematics, 5(3), 102-114.

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