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I'm meant to prove the statement above and I chose to do so by thinking about the equation modulo $4$ and residue classes. My simplified process is below:

$c^2$ is either the residue class $1$ (if odd) or $0$ (if even). After being multiplied by $3$, it is either the residue class $3$ or $0$.

Similarly, $a^2$ and $b^2$ are either the residue classes $1$ (if odd) or $0$ (if even). Because of this, $a^2+b^2$ can never equal $3$ (which is $3c^2$'s residue class if it's odd), meaning that $a^2$ and $b^2$ must both be even so that $0 + 0 = 0$.

Where I'm lost is where to go from here. My instinct is to say that, since we ignoring the cases where $a$, $b$ and $c$ are zero, then the case above is invalid because $0$ is part of the residue class $0$, but that doesn't seem correct to me. Does anyone have any advice on where to go from here (or if I missed something in my proof?)

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O. Bates
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2 Answers2

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You proved correctly that if $a,b,c\in\mathbb N$ are such that $a^2+b^2=3c^2$, then $a$, $b$ and $c$ must all be even numbers. So, you have $a=2m$, $b=2n$, and $c=2p$. But then $a^2+b^2=3c^2$ means that $m^2+n^2=3p^2$. And now you can start all over again. This is impossible, since you can't divide a positive integer by $2$ again and again and to keep always getting positive integers.

José Carlos Santos
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Hint If $a,b,c$ all all even, cancel a 2 and repeat.

N. S.
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