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Let $k$ and $n$ be positive integers. Show that $$(k+1)^2k^2(n+1)^4-2k(k+1)n(n+1)^2(2kn+k+1)+n^2(k+1)^2$$ is a perfect square if and only if $k=n$.

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    If it helps, you can reduce the polynomial modulo $k$ and $n$: $$F(k,n)\equiv n^2 \pmod k$$ $$F(k,n) \equiv (k^2+k)^2 \pmod n.$$ You can also reduce it modulo some other things, like $k^2+k$...2011-07-17
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    Two of the terms have $(k+1)^2$ so the third must as well. Where does that come from?2011-07-17
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    @Ross, why must the third term have $(k+1)^2$? What if $k+1$ is a square?2011-07-17

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