Pythagorean Triples

A Pythagorean triple is formed by three natural numbers that determine the lengths of the sides of a right triangle.

In a right triangle, according to the Pythagorean theorem, the sum of the squares of the two smaller numbers is equal to the square of the largest number.

\(\displaystyle c^2=a^2+b^2\)

If a, b and c have no divisor in common other than 1, it is called a primitive Pythagorean triple.

The smallest and most famous Pythagorean triple is \((3,\ 4,\ 5)\). It is primitive because the three natural numbers only have 1 as a divisor in common.

Other examples of other small primitive Pythagorean triples are \((5,\ 12,\ 13)\) and \((8,\ 15\ 17)\).

Examples of non-primitive Pythagorean triples are \((15,\ 20,\ 25)\) with \(5\) as a common divisor or \(\displaystyle (15,\ 36,\ 39)\) with the common divisor \(3\).

Generation of Pythagorean triples

The following three formulas produce a Pythagorean triple \((a,\ b,\ c)\) for any \(m,\ n\).

\(\displaystyle a=m^2-n^2\)
\(\displaystyle b=2mn\)
\(\displaystyle c=m^2+n^2\)

The Pythagorean triple is primitive if and only if \(m\) and \(n\) are coprime and not both odd.

Further information on the derivation of the formulas can be found at Wikipedia.