Pythagorean Triples Calculator

Three positive integers a, b, c with a squared plus b squared equals c squared. The triple is primitive if a, b, c have no common factor greater than 1.

Use Euclid's formula with m greater than n, both positive integers, with greatest common divisor one and opposite parity. Every primitive Pythagorean triple arises this way exactly once.

Because (m squared minus n squared) squared plus (2 m n) squared equals (m squared plus n squared) squared is a polynomial identity, derived from the factorization of c squared minus a squared.

The smallest primitive triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a triple, such as (6, 8, 10), is also a Pythagorean triple.

Yes. Euclid's formula a = m² − n², b = 2mn, c = m² + n² generates a triple for any integers m greater than n, and there are infinitely many such pairs, so the triples never run out.