A perfect square problem [solution]

The puzzle was to find all positive integers $n$ (with proof) for which $$n^4 + n^3 + n^2 +n + 1$$ is a perfect square.

Solution

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One liner!

For $n \gt 3$, we have that

$$(2n^2+n)^2 \lt 4(n^4 + n^3 + n^2 +n + 1) \lt (2n^2 + n + 1)^2$$

So verify for $n=1,2,3$ and we get the answer to be $n=3$.