integer right triangles

[Joe Horn]

(10-15-2019 02:34 AM)vanLudwig Wrote:  Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. The formula states that the integers
$$a=m^{2}-n^{2}, b=2mn, c=m^{2}+n^{2}$$
form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and not both odd.

A bit of related fun: The sequence of Pell Numbers (1, 2, 5, 12, 29, 70, 169, ...) is defined recursively as P(1)=1, P(2)=2, and P(n)=2*P(n-1)+P(n-2). Try consecutive pairs of Pell Numbers (e.g. 1 and 2, or 2 and 5) for m and n in the above formulas, and as they get bigger, you'll see why the Pythagorean Triples they yield are called "almost-isosceles primitive Pythagorean triangles".

Spoiler Alert: the above is an immediate consequence of the matrix definition of Pell Numbers:



Just take the determinant of both sides. Voilà!