Here's a fun little exercise I thought of. The Pythagorean triples are those triples $(a,b,c)$ of positive integers such that $a^2 + b^2 = c^2$) Some examples are (3,4,5), (6,8,10), and (5,12,13). In fact there is an infinite amount of these. For convenience we shall always list them such that $a < b < c$.

If we specifically we look at the ones where the first number is a prime we find an interesting pattern. Let's take a look at the first few!
(3,4,5)
(5,12,13)
(7,24,25)
(11,60,61)
It seems as though if the first number is a prime, then the last number is just one more than the middle one!

Prove this.