Structure and spectra of the components of primitive Pythagorean triples and Fermat’s last theorem

J. V. Leyendekkers, A. G. Shannon and C. K. Wong
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 15, 2009, Number 4, Pages 13–22
Full paper (PDF, 204 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, NSW 1465, Australia

C. K. Wong
Warrane College, The University of New South Wales
Kensington, NSW 1465, Australia

Abstract

Certain characteristics of Pythagorean triples are analysed using Integer Structure via the modular ring, Z₄. The fact (as shown by Fermat’s Last Theorem) that all the components of a triple cannot simultaneously be an even power n (with ½n even) is illustrated via the spectra of the right-end-digits of the components.

Keywords

• Pythagorean triples
• Right end digits
• Integer structure analysis
• Modular rings

AMS Classification

• 11D41
• 11A07

References

1. Aczel, Amir D. 1997. Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. New York: Delta.
2. Artin, Emil. Otto Schreier. Algebraische Konstruktion reeler Körper. Abhandlungen aus dem Mathematichen Seminar der Universität Hamburg. 5, 1927:85-99.
3. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9, 2007.
4. Zassenchaus, H., Emil Artin. His Life and Work. Notre Dame Journal of Formal Logic. 5, 1964: 1-9.