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Post Closed as "Duplicate" by Pace Nielsen, Martin Sleziak, Max Horn, Alexey Ustinov, user44191
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Integer solutions to $x^2 + x + 1 = y^z$?

In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on the case $x=18$, $y=7$, $z=3$. Does anyone know more about integer solutions to this equation?