<s>[INST] It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square Prove that it is impossible for three consecutive squares to sum to another perfect square.
I have tried for the three numbers $x-1$, $x$, and $x+1$.
 [/INST] The smallest solution you may think of:
Note that $(x-1)^2+x^2+(x+1)^2=3x^2+2\equiv 2\pmod{3}$, but $$y^2\equiv 0,1\pmod{3}$$for all integers $y$. Therefore, $(x-1)^2+x^2+(x+1)^2$ can never be a perfect square.
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