Analysis of Diophantine properties using modular rings with four and six classes

J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 3, 1997, Number 2, Pages 61–74
Full paper (PDF, 398 Kb)

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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

A modular ring ℤ4 is described, and used together with a modular ring ℤ6 and Pythagorean-triple grid, described earlier, to analyse various diophantine properties and explain why the area of a Pythagorean triangle can never be a square.

References

  1. 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.
  2. J. V. Leyendekkers and J. M. Rybak, The generation and analysis of Pythagorean triples within a two-parameter grid. International Journal of Mathematical Education in Science and Technology, 26, 6, 1995, 787-93.
  3. J. Hunter, Number Theory, Oliver and Boyd, Edinburgh, 1969

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  3. Leyendekkers, J., & Shannon, A. (2002). Integer structure and constraints on powers within the modular ring ℤ4 – Part I: Even powers. Notes on Number Theory and Discrete Mathematics, 8(2), 41-57.
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  5. Leyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1998). The characteristics of primes and other integers within the modular ring Z4 and in class ̅1. Notes on Number Theory and Discrete Mathematics, 4(1), 1-17.

Cite this paper

Leyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1997). Analysis of Diophantine properties using modular rings with four and six classes. Notes on Number Theory and Discrete Mathematics, 3(2), 61-74.

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