Pythagorean Triples

[Dieter]

(02-09-2017 05:42 AM)Eddie W. Shore Wrote:  Pythagorean triples can be generated with three arbitrary positive integers M, N, and K with the following criteria:

1. M > N
2. M and N are coprime. That is, gcd(M, N) = 1 (gcd, greatest common denominator)

Eddie, M and N do not have to be coprime (which can be easily shown). If they are, K=1 produces the smallest possible Pythagorean triple. But this is not required for generating such triples in general. Any M > N will do. If the GCD of M and N is G instead of 1 the result is the same as if you would use G²*K instead of K in your formula.

Example: M=10 and N=20 yields 300² + 400² = 500².
Which is 10² times the result of M=1 and N=2, leading to 3² + 4² = 5².

So the GCD condition can be dropped. If your goal is generating primitive Pythagorean triples, a third condition has to be added: M and N must not be both odd, it has to be one odd and one even value, cf. the Wikipedia article you linked to.

Dieter