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?
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$\begingroup$ I think this question (or variants thereof ) has been asked before on this site. I will try to find some of the questions. $\endgroup$– Geoff RobinsonCommented Dec 6, 2023 at 10:13
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2$\begingroup$ Look at question 206931 and its answers. The second answer refers to the Nagell-Ljunggren equation, and its solution by Ljunggren- was writing my comment as @GH from MO's comment came in. $\endgroup$– Geoff RobinsonCommented Dec 6, 2023 at 10:19
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$\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$– GH from MOCommented Dec 6, 2023 at 10:20
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$\begingroup$ Indeed, why not? $\endgroup$– Maarten HavingaCommented Dec 7, 2023 at 7:26
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2 Answers
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The only solution is $x=18$, $y=7$, $z=3$. See my response at What is prime power of this equation of p?
answered Dec 6, 2023 at 10:23
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You should definitely take a look at this previous answer of mine:
answered Dec 6, 2023 at 21:12