Can we find three odd numbers satisfying this relation?

Question

If $a^2+b^2=c^2$ and $(a, b, c) \in \mathbb N^3$ are natural numbers. Can we find three odd numbers satisfying this relation?

Answer 1

The square of an odd number is still odd, and the square of an even number is even. So a number $x$ is odd if and only if its square $x^2$ is odd.

We deduce $a^2 + b^2$ is even, so $c^2$ is even, and $c$ is even. Contradiction.

Answer 2

Let $ a, b $ be odd. Then $ a^2, b^2 $ are as well. Thus, $ a^2 + b^2 $ is even, which means that $ c^2 $ is even. Thus, $ c $ cannot be odd with $ a, b $ even.