Pythagorean triple

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.

For example:

              a          b          c
              3          4          5
              6          8         10
              5         12         13
              9         12         15
              8         15         17
              7         24         25

If (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

is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every primitive triple (possibly after exchanging a and b) arises in this fashion from a unique pair of coprime numbers m > n. This shows that there are infinitely many primitive Pythagorean triples.

Fermat's Last Theorem states that non-trivial triples analogous to Pythagorean triples but with exponents higher than 2 don't exist.