Odd powers as sums of squares

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 1̅₄ of the Modular Ring Z₄. 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 (a₁, b₁). A general equation is given for predicting the square couples of pⁿ 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 ̅2₄ where there are no powers at all.

AMS Classification

• 11A41
• 11A07

References

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