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?

  • 1
    $\begingroup$ Where do you see a pair when $a,b,c$ are three numbers? $\endgroup$ – Hagen von Eitzen Jul 17 '16 at 9:08
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    $\begingroup$ How can two odd numbers add up to another odd number $\endgroup$ – Archis Welankar Jul 17 '16 at 9:09
  • $\begingroup$ That is the question actually.. It is either possible or not possible.. I am asking for if it is possible $\endgroup$ – danny Jul 17 '16 at 9:12
  • $\begingroup$ The title says one thing, the body says another. $\endgroup$ – barak manos Jul 17 '16 at 9:38

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.


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