J. V. Leyendekkers and A. G. Shannon

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

Volume 17, 2011, Number 2, Pages 40—46

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006, Australia*

A. G. Shannon

*Faculty of Engineering & IT, University of Technology Sydney
2007 Australia*

### Abstract

Structural analysis (via the modular rings Z_{4}, Z_{6}) shows that the Perimeters, Pr, of primitive Pythagorean Triples (pPts) do not belong to simple functions. However, the factors *x*, (*x+y*) of the perimeter do, and the number of pPts in a given interval can be estimated from this. When *x* is prime, the series for (*x+y*) is complete and the associated pPts are one third of the total. When *x* is composite, members of the series for (*x+y*) are invalid when common factors with *x* occur. These members are not associated with pPts. When 3|(*x+y*), Pr ∈ ̅3_{6}, while if 3 ∤(*x+y*), Pr ∈ { ̅1_{6}, ̅3_{6}}. Class ̅3_{6} dominates in the distribution.

### Keywords

- Primitive Pythagorean triples
- Modular rings
- Primes
- Composites

### AMS Classification

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

### References

- Lehmer, D.N. 1900. Asymptotic Evaluation of Certain Totient Sums.
*American Journal of Mathematics.*22: 294-335. - 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. Equations for Primes Obtained from Integer Structure.
*Notes on Number Theory & Discrete Mathematics.*16(3): 1-10. - Leyendekkers, J.V., A.G. Shannon. 2011. The Number of Primitive Pythagorean Triples in a Given Interval.
*Notes on Number Theory & Discrete Mathematics*In press.

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## Cite this paper

APALeyendekkers, J. V., and Shannon, A. G. (2011). Structure analysis of the perimeters of primitive Pythagorean triples. Notes on Number Theory and Discrete Mathematics, 17(2), 40-46.

ChicagoLeyendekkers, JV, and AG Shannon. “Structure Analysis of the Perimeters of Primitive Pythagorean Triples.” Notes on Number Theory and Discrete Mathematics 17, no. 2 (2011): 40-46.

MLALeyendekkers, JV, and AG Shannon. “Structure Analysis of the Perimeters of Primitive Pythagorean Triples.” Notes on Number Theory and Discrete Mathematics 17.2 (2011): 40-46. Print.