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)**

## Details

### 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 ℤ_{4} we illustrate how the underlying Class structures of the integers justifies this constraint.

### AMS Classification

- 11C08
- 11D41

### References

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Hunter, J. 1964. Number Theory. Edinburgh: Oliver and Boyd. - 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.
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_{8}. Notes on Number Theory & Discrete Mathematics. Vol.5(3): 102-114. - 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.
- Leyendekkers, J.V. & Shannon, A.G. In press. Integer Structure and Constraints on Powers within the Modular Ring ℤ
_{4}. Notes on Number Theory & Discrete Mathematics. - Niven, I. & Zuckerman, H.S. 1966. An Introduction to the Theory of Numbers. Second Edition. New York: Wiley.
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## Related papers

- 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.
- Leyendekkers, J., & Shannon, A. (2002). Integer structure and constraints on powers within the modular ring ℤ
_{4}– Part I: Even powers.*Notes on Number Theory and Discrete Mathematics*, 8(2), 41-57. - Leyendekkers, J., & Shannon, A. (2002). Integer structure and constraints on powers within the modular ring ℤ
_{4}– 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.