Pythagorean Triples and Euclid’s Formula

The Pythagorean theorem states that

a^2 + b^2 = c^2

When $a, b, c$ are positive integers that satisfy this relation, the triple $(a, b, c)$ is called a Pythagorean triple.
For example, $(3, 4, 5)$ is a well-known Pythagorean triple.

Euclid’s Formula

One systematic way to generate Pythagorean triples is through Euclid’s formula.

Choose two natural numbers $m, n$ with $m > n$ .
Define the triple as follows:

a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2

Then $(a, b, c)$ will be a Pythagorean triple.

Proof

By direct calculation,

(m^2 - n^2)^2 + (2mn)^2 = m^4 + 2m^2n^2 + n^4 = (m^2 + n^2)^2

which shows that $(a, b, c)$ always satisfies

a^2 + b^2 = c^2