J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 13, 2007, Number 1, Pages 16–24

**Full paper (PDF, 113 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, 1465, &
Raffles KvB Institute Pty Ltd, North Sydney, NSW 2060, Australia
*

### Abstract

Odd integers raised to an odd power *n* can equal a sum of squares only if the integers are in Class 1̅_{4} of the Modular Ring Z_{4}. The primes raised to an odd power are unique in that they only have (*n* − 1) square couples. These couples depend on the single couple (*a*_{1}, *b*_{1}). A general equation is given for predicting the square couples of *p ^{n}* and odd-powered composites. Even integers raised to an odd power have no primitive solutions for such square couples, because the sum of two odd squares falls in Class ̅2

_{4}where there are no powers at all.

### AMS Classification

- 11A41
- 11A07

### References

- John H Conway & Richard K Guy,
*The Book of Numbers*. New York: Copernicus, 1996. - J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, The Characteristics of Primes and Other Integers within the Modular Ring Z
_{4}and in Class 1̅_{4}.*Notes on Number Theory & Discrete Mathematics*. 4 (1) (1998): 1-17. - J.V. Leyendekkers & A.G. Shannon, The Row Structure of Squares in Modular Rings.
*Notes on Number Theory & Discrete Mathematics*. 10 (1) (2004): 1-11. - Neal H. McCoy,
*The Theory of Numbers*. New York: Macmillan, 1965, Ch.2. - W.G. Nowak. On Sums and Differences of Two Relative Prime Cubes.
*Analysis*. 15 (1995): 325-341.

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## Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2007). Odd powers as sums of squares. *Notes on Number Theory and Discrete Mathematics*, 13(1), 16-24.