Using integer structure to solve Diophantine equations

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 12, 2006, Number 3, Pages 10–19
Full paper (PDF, 85 Kb)

Details

Authors and affiliations

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

A. G. Shannon
Warrane College, The University of New South Wales, Kensington, 1465,
& KB Institute of Technology, North Sydney, NSW 2060, Australia

Abstract

Diophantine equations {ax + by = c; a, b, c ∈ ℤ} are classified according to parity constraints. Various types, so classified, are solved with the theory of integer structure, via the modular ring Z₄. The simplest forms are those where one of the variables is confined to a single class. However, the more complex equations have solutions that follow regular (x, y) class patterns. The famous Diophantine equation in Fermat’s Last Theorem is discussed in terms of the factor structure of the sum of two powers.

AMS Classification

• 11A41
• 11A07

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