A new approach to generate all Pythagorean triples

Anthony Overmars; Lorenzo Ntogramatzidis; Sitalakshmi Venkatraman; Anthony Overmars; Lorenzo Ntogramatzidis; Sitalakshmi Venkatraman

doi:10.3934/math.2019.2.242

AIMS Mathematics

2019, Volume 4, Issue 2: 242-253. doi: 10.3934/math.2019.2.242

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Research article

A new approach to generate all Pythagorean triples

1.
Department of Information Technology, Melbourne Polytechnic, VIC, Australia
2.
Department of Mathematics and Statistics, Curtin University, Perth (WA), Australia

Received: 08 December 2018 Accepted: 14 February 2019 Published: 14 March 2019

This paper revisits the topic of Pythagorean triples with a different perspective. While several methods have been explored to generate Pythagorean triples, none of them is complete in terms of generating all the triples without repetitions. Indeed, many existing methods concentrate on generating primitive triples but do not cater to non-primitives. By contrast, the approach presented in this paper to parameterise the Pythagorean triples generates all of the triples in a unique way, i.e., without repetitions. We also explore the relation of this new parameterisation with the Pythagorean family of odd triples and with the Platonic family of even triples.
- Pythagorean triples,
- Diophantine equations,
- Euclidean triples,
- Platonic triples,
- right angle triangles
Citation: Anthony Overmars, Lorenzo Ntogramatzidis, Sitalakshmi Venkatraman. A new approach to generate all Pythagorean triples[J]. AIMS Mathematics, 2019, 4(2): 242-253. doi: 10.3934/math.2019.2.242

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Abstract

This paper revisits the topic of Pythagorean triples with a different perspective. While several methods have been explored to generate Pythagorean triples, none of them is complete in terms of generating all the triples without repetitions. Indeed, many existing methods concentrate on generating primitive triples but do not cater to non-primitives. By contrast, the approach presented in this paper to parameterise the Pythagorean triples generates all of the triples in a unique way, i.e., without repetitions. We also explore the relation of this new parameterisation with the Pythagorean family of odd triples and with the Platonic family of even triples.

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