Three positive integers a, b, c such that a2 + b2 = c2 are said to form a Pythagorean triple. The name comes from the Pythagorean Theorem, which states that any right triangle with integer side lengths yields a Pythagorean triple. The converse is also true: every Pythagorean triple determines a right triangle with the given side lengths.
a b c 3 4 5 6 8 10 5 12 13 9 12 15 8 15 17 7 24 25If (a,b,c) is a Pythagorean triple so is (da,db,dc) for any positive integer d. A Pythagorean triple is said to be primitive if a, b and c have no common divisor. The triangles described by non-primitive Pythagorean triples are always proportional to the triangle described by a smaller primitive Pythagorean triple.
If m > n are positive integers, then
- a = m2 − n2,
- b = 2mn,
- c = m2 + n2
Fermat's Last Theorem states that non-trivial triples analogous to Pythagorean triples but with exponents higher than 2 don't exist.