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.

A good starting point for exploring Pythagorean triples is to recast the original equation in the form:

a² = (c − b)(c + b)

It is interesting to note that there are more than one primitive Pythagorean triple with the same lowest integer, the first example is for 20, which is the lowest integer of two primitive triples: 20 21 29 and 20 99 101.

By contrast the number 1229779565176982820 is the lowest integer in exactly 15386 primitive triples, the smallest and largest triples it is part of are:

1229779565176982820
1230126649417435981
1739416382736996181

and

1229779565176982820
378089444731722233953867379643788099
378089444731722233953867379643788101.

For the curious, consider the prime factorisation of 1229779565176982820 = 2² * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47. The number of prime factors is related to the large number of primitive Pythagorean triples. Note that there are larger integers that are the lowest integer in an even greater number of 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.

External links

• http://mathworld.wolfram.com/PythagoreanTriple.html has an extensive discussion of Pythagorean triples.
• http://www.math.clemson.edu/~rsimms/neat/math/pyth/ provides a Javascript calculator for the (m² − n², 2mn, m² + n²) formula, and shows how to derive the formula.