The number of primitive Pythagorean triples in a given interval

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 1, Pages 49–57
Full paper (PDF, 184 Kb)

Details

Authors and affiliations

J. V. Leyendekkers

Faculty of Science, The University of Sydney
Sydney, NSW 2006, Australia

A. G. Shannon

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

Abstract

Integer structure analysis within the framework of modular rings is used to show that the formation of primitive (or “reduced”) Pythagorean triples depends on certain characteristics with these rings. Only integers in the Class 1̅₄ of the modular ring Z₄ can produce primitive Pythagorean triples. Of these, a prime produces only one primitive Pythagorean triple, while composites produce the same number of primitive Pythagorean triples as their factors, provided the factors are square-free or are not elements of 3̅₄ . Class 1̅₄ integers were converted to the equivalent Z₆ classes in order to isolate those divisible by 3. The numbers of primitive Pythagorean triples in various ranges were estimated and compared with the elder Lehmer’s estimates. The results provide a neat link between the number of primitive Pythagorean triples and the number of primes in the given interval. It was also shown why the major component of a primitive Pythagorean triple is the only component which cannot have 3 as a factor.

Keywords

• Primitive Pythagorean triples
• Modular rings
• Primes
• Composites

AMS Classification

• 11A41
• 11A07
• 11B39
• 11C99

References

1. Hall, A. Genealogy of Pythagorean Triads. Mathematical Gazette. Vol. 54, 1970, 377–379.
2. Lehmer, D.N. Asymptotic Evaluation of Certain Totient Sums. American Journal of Mathematics. Vol. 22, 1900, 294–335.
3. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No. 9, 2007.
4. Leyendekkers, J.V., A.G. Shannon. Why 3 and 5 are always Factors of Primitive Pythagorean Triples. International Journal of Mathematical Education in Science & Technology. Vol. 42, 2011, 102–105.