Generalizations on the well known Pythagorean triple divisors, and
Let the ordered triple be a primitive Pythagorean triple where is a positive integer and is the odd leg. Then there exists relatively prime odd positive integers and where such that
Since and are relatively prime, and for some odd positive integers and . Then equation (20) becomes
If is a divisor of then where is the integer . Thus, we can rewrite equations (22) as
Note: If a prime divides then divides .
Therefore, from equation (23) and Fermat's little theorem, if prime then divides . And if prime then divides . That is, since and are pairwise relatively prime, divides exactly one of or and divides exactly one of or
then divides the even side .
Therefore divides .