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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?
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.
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.