One of the most well known exponential diophantine equations is $(x^n-1)/(x-1)=y^p$, where $x,n,y,p$ are positive Integers with $n>2$, $x>1$, and $p$ prime. Some solutions are obtained with $x=3, n=5$, $x=7,n=4$, and also with $x=18,n=3$. Are ALL solutions classified? Probably not. That said, the case I am most interested is when $x$ is prime, because It is equivalent to the problem of determining all perfect powers among numbers of form $\sigma{(p^m)}$, $p$ prime, $m$ natural number. Is more known in this case besides the two solutions given above? Can the full classification be done?
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4$\begingroup$ This equation is known as the Nagell-Ljunggren equation. This should let you do a literature search. In particular, as you guess, it is wide open still. This is the most recent paper I found surveying known results. I don't think much more can be said in the case $x$ is prime. $\endgroup$Wojowu– Wojowu2025-12-07 23:26:59 +00:00Commented Dec 7 at 23:26
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