# Pythagorean triple

In a Pythagorean triple there are three numbers: a, b and c and they are related to each other:

$\Bigg \{ \begin{array}{ll} a < b < c \\ a^2 + b^2 = c^2 \end{array}$

Represented in a Cartesian plane, a and b represent the two sides of a right-angled triangle while c is the hypotenuse.

## Non-primitive Pythagorean triple

If a, b and c are coprime the triad is called primitive Pythagorean triple. Multiplying all three numbers by the same factor, we obtain a Pythagorean triplet, which takes the name of non-primitive Pythagorean triplet.

## Generating a triple

Euclid’s formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. The formula states that the integers

$\Bigg \{ \begin{array}{ll} a = m^2-n^2 \\ b = 2mn \\ c = m^2+n^2 \end{array}$

form a Pythagorean triple. a, b e c may form a non-primitive Pythagorean triple if m and n are not coprime and not both odd.