These numbers are called Pythagorean triples, as they satisfy Pythagoras Theorem. More generally, they are solutions of the Diophantine equation $a^2 + b^2 = c^2$. Here are a few more:
- $8^2 + 15^2 = 17^2$
- $24^2 + 7^2 = 25^2$
- $40^2 + 9^2 = 41^2$
A larger list can be found on Wikipedia. There are a lot of formulae to generate Pythagorean triples. According to Euclid's formula, the numbers
$$a = m^2 - n^2, b = 2nm, c = m^2 + n^2$$
form a pythagorean triple $a^2 + b^2 = c^2$ for all combinations of $m$ and $n$.
Note that multiples of $a, b, c$ i.e. $ka, kb, kc$ also satisfy the equation. When $a, b, c$ have a highest common factor of 1, the triple is called a primitive pythagorean triple.
You can find out more about pythagorean triples in the mentioned links.