Pythagorean Triple - Generating A Triple

Generating A Triple

Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with m > n. The formula states that the integers

form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and m − n is odd. If both m and n are odd, then a, b, and c will be even, and so the triple will not be primitive; however, dividing a, b, and c by 2 will yield a primitive triple if m and n are coprime.

Every primitive triple (possibly after exchanging a and b) arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of a and b to m and n from Euclid's formula is referenced throughout the rest of this article.

Despite generating all primitive triples, Euclid's formula does not produce all triples. This can be remedied by inserting an additional parameter k to the formula. The following will generate all Pythagorean triples (although not uniquely):

where m, n, and k are positive integers with m > n.

That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra and verifying that the result coincides with c2. Since every Pythagorean triple can be divided through by some integer k to obtain a primitive triple, every triple can be generated (perhaps not uniquely) by using the formula with m and n to generate its primitive counterpart and then multiplying through by k as in the last equation.

Many formulas for generating triples have been developed since the time of Euclid.