Prove that any quadratic of the form $x^4+(2k^2)x^2+2k^4$ for any nonzero integer $k$ is never a perfect square for all integers $x$.
I thought about proving this by induction. We first prove that $x^4+2x^2+2$ is not a perfect square for all integers $x$ (which is true since $x^4+2x^2+2 = (x^2+1)^2+1$) and then proceed with the inductive hypothesis. Is there an easier way of solving this?
nonzeropart. Maybe related: A quadratic polynomial getting square values in consecutive points. – dxiv Dec 24 '16 at 03:53