To prove that $2^{3n}+2^n +1$ is not a perfect square.

Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$.

I attempted this question and failed in two different ways.

1) I considered a polynomial $p(x) = x^3+ x + 1 - m^2$ (for some natural $m$) and factorized the polynomial assuming $2^n$ is a root. I, then, tried substituting some numbers to get a contradiction in divisibility (since $x - 2^n$ is a factor). Alas, all of that was in vain.

2) I wrote $2^{3n} + 2^n + 1 = m^2$ and thus $(m-1)(m+1) = 2^n(2^{2n} + 1)$. So one can conclude that if $xy = 2^{2n} + 1$ such that $(x,y) = 1$, then one of the $m-1$ or $m+1$ must be $2^{n-1}x$ and the other must be $2y$ (this follows from the fact that $m$ is odd and $(m-1,m+1) = 2$). I am stuck at this point.

I would appreciate a hint in either direction. I am hoping that 1) will work out. It will teach me a new proofing (err... proving) technique.

Thanks!

Hint: Consider 2 cases. If $n$ is odd, take $\pmod{3}$. If $n$ is even, bound between 2 squares.