Powers as a difference of squares: The effect on triples

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 8, 2002, Number 3, Pages 95–106
Full paper (PDF, 519 Kb)

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Authors and affiliations

J. V. Leyendekkers
The University of Sydney
NSW 2006, Australia

A. G. Shannon
Warrane College, The University of New South Wales, 1465, &
KvB Institute of Technology, North Sydney, 2060, Australia

Abstract

All powers equal a difference of squares so that triples may be expressed as the sum of three squares which equal the sum of another three squares. However, when n > 2 integer values for all the components should be impossible according to the work which peaked with Wiles. By utilizing the properties of the Modular Ring ℤ4 we illustrate how the underlying Class structures of the integers justifies this constraint.

AMS Classification

  • 11C08
  • 11D41

References

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  4. 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 & Discrete Mathematics. Vol.3(2): 61-74.
  5. Leyendekkers, J.V. & Shannon, A.G. 1999. Analyses of Row Expansions within the Octic ‘Chess’ Ring ℤ8. Notes on Number Theory & Discrete Mathematics. Vol.5(3): 102-114.
  6. Leyendekkers, J.V. & Shannon, A.G. 2001. Expansion of Integer Powers from Fibonacci’s Odd Number Triangle. Notes on Number Theory & Discrete Mathematics. Vol.7(2): 48-59.
  7. Leyendekkers, J.V. & Shannon, A.G. In press. Integer Structure and Constraints on Powers within the Modular Ring ℤ4. Notes on Number Theory & Discrete Mathematics.
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Related papers

  1. Leyendekkers, J. V., & Shannon, A. G. (2003). Some characteristics of primes within modular rings. Notes on Number Theory and Discrete Mathematics, 9(3), 49-58.
  2. 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.
  3. Leyendekkers, J., & Shannon, A. (2002). Integer structure and constraints on powers within the modular ring ℤ4 – Part II: Odd powers. Notes on Number Theory and Discrete Mathematics, 8(2), 58-66.

Cite this paper

Leyendekkers, J., & Shannon, A. (2002). Powers as a difference of squares: The effect on triples. Notes on Number Theory and Discrete Mathematics, 8(3), 95-106.

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