Do simple equations $x^2+x+1=7^n$ have infinitely or finitely many solutions? In fact, what about the general equation $x^2+x+1=p^n$ where $p$ is a prime congruent to $1$ $\pmod 3$?
Are there any more solutions $(x,n)$ to $x^2+x+1=7^n$ besides $(2,1)$ and $(18,3)$? I would expect no but I am not sure?
I know the case $x^2+1=5^n$ has finitely many solutions $(x,n)$ (and $(2,1)$ is the only solution) due to Catalan's theorem on consecutive perfect powers.