<s>[INST] Is there a perfect square that is the sum of $3$ perfect squares? This is part of a bigger question, but it boils down to:
Is there a square number that is equal to the sum of three different square numbers?
I could only find a special case where two of the three are equal? https://pir2.forumeiros.com/t86615-soma-de-tres-quadrados (in portuguese).
Any clue?
 [/INST] We know that $(n+1)^2-n^2=2n+1$, so pick your favorite Pythagorean triple $a^2+b^2=c^2$ with $c$ odd.  Let $c^2=2n+1, n=\frac {c^2-1}2$ and 
$$a^2+b^2+n^2=(n+1)^2$$
If you pick a triple with $c$ even, we can use $(n+2)^2-n^2=4n+4$, so we can let $n=\frac {c^2-4}4$ and have $$a^2+b^2+n^2=(n+2)^2$$
If $c$ is even, $c^2$ is divisible by $4$, so the division will come out even.
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