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 Z_{4}. 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

### References

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## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2006). Using integer structure to solve Diophantine equations. *Notes on Number Theory and Discrete Mathematics*, 12(3), 10-19.