Using completing the square and factoring method I could show that the Diophantine equation $x^2+x+1=y^n$, where $x,y$ are odd positive and $n$ is even positive integers, does not have solution, but I could not show that for odd positive $x,y$ and odd $n>1$ the equation does (does not) have solution.

I already asked the above question of some expert and I received good information about such equation, but how it will be if we assume $x,y$ were "odd prime" numbers?

Thank you for your contribution.

P.S. I already put it here, but did not get some useful suggestion.