Problem #PRU-100645

Problem

A triple of natural numbers $a,b,c$ such that $a^2+b^2=c^2$ is called a Pythagorean triple. There are some small Pythagorean triples that are well-known, like $3,4,5$ and $5,12,13$. Let us have a look at the latter one. We can notice an interesting thing: not only $5^2+{12}^2={13}^2$, but also $5^2=25=12+13$ and $13-12=1$. Use that as an inspiration to find an idea of how to generate some more Pythagorean triples. Check if they are correct by plugging them into the equation $a^2+b^2=c^2$.