J. V. Leyendekkers and A. G. Shannon

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

Volume 15, 2009, Number 3, Pages 14–20

**Full paper (PDF, 163 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, Australia*

### Abstract

The modular ring *Z*_{4} was used to analyse the structure of the integer, *N*, obtained from *x ^{n} − y^{n}*,

*x*,

*y*,

*n*odd. The constraints on

*x*and

*y*associated with the probability of

*x*(

^{n}− y^{n}= N = z^{n}*z*even) were explored. When

*n*∈ ̅3

_{4}(

*n*= 3, 7, 11, 15, …) the structure of

*N*is 4

*r*

_{0}(4

*r*

_{3}+ 3) that is ̅0

_{4}× ̅3

_{4}. When

*n*∈ ̅1

_{4}(

*n*= 5, 9, 13, 17, …) the structure of

*N*is 4

*r*

_{0}(4

*r*

_{1}+ 1) that is ̅0

_{4}× ̅1

_{4}. The row structures and right-end-digit patterns of the rows of (

*x*

^{3}−

*y*

^{3}) and

*z*

^{3}were compared and shown to be incompatible, as expected.

### Keywords

- Primes
- Composites
- Modular rings
- Right-end digits
- Integer structure

### AMS Classification

- 11A41
- 11A07

### References

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## Related papers

## Cite this paper

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.