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

**Download 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

- Clarke, J.H., A.G. Shannon.1983. A combinatorial approach to Goldbach’s conjecture.
*Mathematical Gazette*. 67: 44–46. - Cornell, G., J.H. Silverman, G. Stevens (eds.) 1997.
*Modular Forms and Fermat’s Last**Theorem*. Berlin: Springer. - Faltings, Gerd. 1995. The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles.
*Notices of the American Mathematical Society*. 42: 743-746. - Leyendekkers, J.V., A.G. Shannon. (In press) Analysis of Primes Using REDs (Right-End-Digits) and Integer Structure.
*Notes on Number Theory & Discrete Mathematics*. - Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007.
*Pattern Recognition: Modular**Rings and Integer Structure.*North Sydney: Raffles KvB Monograph No.9. - Leyendekkers, J.V., A.G. Shannon. 2007. Modular Ring Class Structures of
*x*±^{n}*y*.^{n}*Notes on Number Theory & Discrete Mathematics*. 13(3): 27-35. - Leyendekkers, J.V., A.G. Shannon, C.K. Wong. 2008. Integer Structure Analysis of the Product of Adjacent Integers and Euler’s Extension of Fermat’s Last Theorem.
*Advanced**Studies in Contemporary Mathematics*. 17: 221-229. - Leyendekkers, J.V., A.G. Shannon, C.K. Wong. 2009. The Spectra of Primes.
*Proceedings of the Jangjeon Mathematical Society*. 12: 1-10. - Leyendekkers, J.V., A.G. Shannon, C.K. Wong. (In press). Structure and Spectra of the Components of of pPts and Fermat’s last Theorem.
*Notes on Number Theory & Discrete**Mathematics*. - Műller-Olm, Markus, Helmut Seidl. 2005. A Generic Framework for Interprocedural Analysis of Numerical Properties. In Chris Hankin, Igor Silveroni (eds).
*Static Analysis*. Berlin: Springer, pp. 235-250. - Wiles, A. 1995. Modular Elliptic Curves and Fermat’s Last Theorem.
*Annals of**Mathematics*. 141: 443-551.

## 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.