Structure analysis of the perimeters of primitive Pythagorean triples

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
Full paper (PDF, 278 Kb)

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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 Z₄, Z₆) 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 ∈ ̅3₆, while if 3 ∤(x+y), Pr ∈ { ̅1₆, ̅3₆}. Class ̅3₆ dominates in the distribution.

Keywords

• Primitive Pythagorean triples
• Modular rings
• Primes
• Composites

AMS Classification

• 11A41
• 11A07
• 11B39
• 11C99

References

1. Lehmer, D.N. 1900. Asymptotic Evaluation of Certain Totient Sums. American Journal of Mathematics. 22: 294-335.
2. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
3. Leyendekkers, J.V., A.G. Shannon. 2010. Equations for Primes Obtained from Integer Structure. Notes on Number Theory & Discrete Mathematics. 16(3): 1-10.
4. 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.