**J. V. Leyendekkers and A. G. Shannon**

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

Volume 16, 2010, Number 4, Pages 6–13

**Full paper (PDF, 168 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

Structural constraints prevent the difference of two odd cubes ever equaling an even cube. This is illustrated from the row structure of the modular ring *Z*_{4}. The critical structure factor is that the rows of integers, *N*^{2}, with 3 | *N*^{2}, follow the triangular numbers, whereas 3 ∤ *N*^{2} rows follow the pentagonal numbers. This structural characteristic is the reason for the importance of primitive Pythagorean triples (in which either the smallest odd component or the even component always has a factor 3).

### Keywords

- Primitive Pythagorean triples
- Triangular numbers
- Pentagonal numbers
- Modular rings
- Cubic triples

### AMS Classification

- 11A41
- 11A07
- 11B39
- 11C99

### References

- Ewell, John A. 1992. On Sums of Triangular Numbers and Sums of Squares.
*American Mathematical Monthly*. 99: 752-757. - Guy, Richard K. 1994. Every Number is Expressible as the Sum of How Many Polygonal Numbers.
*American Mathematical Monthly.*101: 169-172. - Hansen, Rodney T. 1970. Arithmetic of Pentagonal Numbers.
*The Fibonacci Quarterly.*8: 83-87. - Harkleroad, Leon. 2009.
*The Math behind the Music.*New York: Cambridge University Press, p.123. - 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. 2010. Why 3 and 5 are always Factors of Primitive Pythagorean Triples. (Submitted.)
- Leyendekkers, J.V., A.G. Shannon. 2010. Rows of Odd Powers in the Modular Ring
*Z*_{4}. (Submitted.) - Ming, Luo. 1989. On Triangular Fibonacci Numbers.
*The Fibonacci Quarterly.*27: 98-108. - Ollerton, R.L., A.G. Shannon. 2004. Fibonacci’s Odd Number Array.
*Journal of Recreational Mathematics*. 32: 198-204. - Shiro, Ando. 1981. A Note on Polygonal Numbers.
*The Fibonacci Quarterly.*19: 180-183. - Sierpinski, W. 1968. Un theorem sur les nombres triangulaires.
*Elemente der Mathematik.*231: 31-32.

## Related papers

- Leyendekkers, J. V., & Shannon, A. G. (2010). Rows of odd powers in the modular ring
*Z*_{4}.*Notes on Number Theory and Discrete Mathematics*, 16(2), 29-32.

## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2010). Integer structure analysis of odd powered triples: The significance of triangular versus pentagonal numbers. *Notes on Number Theory and Discrete Mathematics*, 16(4), 6-13.