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The formulas $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$ give a Pythagorean triple for integers $m>n>0$, because $(m^2-n^2{)}^2+(2mn{)}^2=(m^2+n^2{)}^2$. They give a primitive Pythagorean triple exactly when $m,n$ are coprime and have opposite integer parity. Every primitive Pythagorean triple arises this way after possibly exchanging the legs; scaling gives the nonprimitive triples.

A Pythagorean triple $(a,b,c)$ whose coordinates have greatest common divisor 1. Its two legs have opposite integer parity, and every Pythagorean triple is a unique positive integer multiple of a primitive Pythagorean triple up to the ordering of its legs.

The equation $a^2+b^2=c^2$. Its positive integer solutions are Pythagorean triples; primitive Pythagorean triples are those with no common factor greater than 1.