Analysis of odd exponent triples within the modular ring ℤ4 using binomial expansions and Fermat reductions

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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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 ℤ4 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

  1. George E. Andrews, Number Therory, W.B. Saunders, Philadelphia, 1971, P.49.
  2. K. Atanassov, General Index Matrices, Comptes rendus de l’Academie Bulgare des Sciences, 40(11), 1987: 15-18.
  3.  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.
  4. J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, Analysis of Diophantine Properties using Modular Rings with Four and Six classes, (submitted).
  5. Alf van der Poorten, Remarks on Fermat’s Last Theorem, Australian Mathematical Society Gazette, 21(5), 1994: 150-159.
  6. Alf van der Poorten, Notes on Fermat’s Last Theorem, Wiley-Interscience, New York, 1996, p.77. 7
  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.
  8. 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

APA

Leyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1997). Analysis of odd exponent triples within the modular ring ℤ4 using binomial expansions and Fermat reductions. Notes on Number Theory and Discrete Mathematics, 3(3), 128-158.

Chicago

Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G. “Analysis of odd exponent triples within the modular ring ℤ4 using binomial expansions and Fermat reductions.” Notes on Number Theory and Discrete Mathematics 3, no. 3 (1997): 128-158.

MLA

Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G. “Analysis of odd exponent triples within the modular ring ℤ4 using binomial expansions and Fermat reductions.” Notes on Number Theory and Discrete Mathematics 3.3 (1997): 128-158. Print.

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