J. V. Leyendekkers, J. M. Rybak and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 3, 1997, Number 3, Pages 128—158

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006, Australia
*

J. M. Rybak

*The University of Sydney, 2006, Australia*

A. G. Shannon

*University of Technology, Sydney, 2007, Australia*

### Abstract

The essential characteristics of integers and the relationships with their powers are explored within the framework of the modular ring ℤ_{4} in order to analyze why odd powered triples with exponents greater than unity cannot exist in integer form. Two methods are given which exploit old expansion and reduction techniques in a new way. By way of conclusion the second method is also illustrated by reference to Pythagorean triples.

### AMS Classification

- 11D41
- 11D99

### References

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- J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, The Anatomy of Even Exponent Pythagorean Triples, Notes on Number Theory and Discrete Mathematics, 2(1), 1996:33-52.
- J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, Analysis of Diophantine Properties using Modular Rings with Four and Six classes, (submitted).
- Alf van der Poorten, Remarks on Fermat’s Last Theorem, Australian Mathematical Society Gazette, 21(5), 1994: 150-159.
- Alf van der Poorten, Notes on Fermat’s Last Theorem, Wiley-Interscience, New York, 1996, p.77. 7
- J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, Integer class properties associated with an integer matrix. Notes on Number Theory and Discrete Mathematics, 1(2), 1995, 53-59.
- J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, The anatomy of Odd- exponent Diophantine triples. Notes on Number Theory and Discrete Mathematics. 3(1), 1997: 34-44.

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## Cite this paper

APALeyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1997). Analysis of odd exponent triples within the modular ring ℤ_{4} using binomial expansions and Fermat reductions. Notes on Number Theory and Discrete Mathematics, 3(3), 128-158.

Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G. “Analysis of odd exponent triples within the modular ring ℤ_{4} using binomial expansions and Fermat reductions.” Notes on Number Theory and Discrete Mathematics 3, no. 3 (1997): 128-158.

Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G. “Analysis of odd exponent triples within the modular ring ℤ_{4} using binomial expansions and Fermat reductions.” Notes on Number Theory and Discrete Mathematics 3.3 (1997): 128-158. Print.