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 $k=n$.
The number is a perfect square if and only if $k=n$
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number-theory
elementary-number-theory
diophantine-equations
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0If 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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2Two 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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2@Ross, why must the third term have $(k+1)^2$? What if $k+1$ is a square? – 2011-07-17