Properties and calculation of pythagorean triples

Abstract

Pythagorean triples, which are sets of three natural numbers (a,b,c) satisfying the equation a2+b2=c2, are among the oldest and most fascinating subjects in number theory. This paper is motivated by the potential of Pythagorean triples as a vehicle for curriculum enrichment at various levels. Students typically meet the Pythagorean theorem early in their careers, but they may not have thought about restricting to integer-valued triples. This step naturally prompts important discussions, such as the difference between finding infinitely many Pythagorean triples and finding all Pythagorean triples. Number-theoretic concepts such as greatest common divisors appear very naturally, and of course there are deeper results including the classification of primitive Pythagorean triples. This paper first proves that any natural number n (with n≥3) is an element of some Pythagorean triple (1 and 2 are not), and the range of its maximal Pythagorean triple is determined, laying a theoretical foundation for finding all its Pythagorean triples. Then the existence conditions for primitive Pythagorean triples are provided. Finally, this paper discusses how the results and methods can be used to support classroom activities, investigations and discussions. Python and MATLAB programs for computing all Pythagorean triples for any natural number are available as supplementary files.