Find all positive integers n such that $n^4 + n^3 + 1$ is a perfect square.

Question

I can’t figure out this number theory question. I think that you should ignore the n^4 part, but I can’t figure out what to do next.

Answer 1

If $4n^4+4n^3+4 = (2k)^2$, then notice that $$(2n^2+n-1)^2\leq(2k)^2\leq(2n^2+n)^2.$$

Things should be easy from here.