Publication Date
7-1-2023
Abstract
There are many right triangles in which all three sides a, b, and c have integer lengths. The triples (a,b,c) formed by such lengths are known as Pythagorean triples. Since ancient times, it is known how to generate all Pythagorean triples: we can enumerate primitive Pythagorean triples -- in which the three numbers have no common divisors -- by considering all pairs of natural numbers m>n in which m and n have no common divisors, and taking a =m2 − n2, b = 2mn, and c = m2 + n2. Multiplying all elements of a triple by the same number, we can get all other Pythagorean triples. The proof of this result -- going back to Euclid -- is technical. In this paper, we provide a commonsense explanation of this result. We hope that this explanation -- which is more general than Pythagorean triples -- can lead to new hypotheses and new results about similar situations.
Comments
Technical Report: UTEP-CS-23-36