J. V. Leyendekkers and A. G. Shannon

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

Volume 13, 2007, Number 3, Pages 27–35

**Full paper (PDF, 355 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, NSW 1465
& RafflesKvB Institute Pty Ltd
North Sydney, NSW, 2060, Australia*

### Abstract

Integer structure theory is used to analyse the factors of sums and differences of two identical powers of two integers, *x* and *y*. For instance, the sum of two identical powers, *m*, cannot form primes when *m* is odd or when *m* is even if the powers are odd and of the form *m*/2. The expanded forms of the factors indicate why the structure acts against the sum ever equalling an identical power. The difference of odd powers can yield primes when *x* − *y* = 1. The difference of even powers cannot yield primes whereas the sum can when *m*/2* ^{n}* is even. However,

*x*

^{2}−

*y*

^{2}can equal a prime when

*x*−

*y*= 1.

### AMS Classification

- 11A41
- 11A07

### References

- Craig, Maurice. The Composition Heresies.
*The Australian Mathematical Society Gazette.*Vol.33, No.4(2006), 265-272. - Dickson, L.E.
*History of the Theory of Numbers*.*Volume 3*. New York: Chelsea, 1952. - Leyendekkers, J.V., A.G. Shannon, Integer Structure Analysis of Primes and Composites from (
*x*^{4}+*y*^{4}).*Notes on Number Theory & Discrete Mathematics*. Submitted. - Rouse Ball, W.W.
*History of Mathematics*. New York: Dover, 1960.

## Related papers

- Leyendekkers, J. V., & Shannon, A. G. (2009). The integer structure of the difference of two odd-powered odd integers.
*Notes on Number Theory and Discrete Mathematics*, 15(3), 14-20.

## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2007). Modular-ring class structures of *x ^{n}* ±

*y*.

^{n}*Notes on Number Theory and Discrete Mathematics*, 13(3), 27-35.