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OPEN
Let $p$ be a prime. Is it true that the equation \[(p-1)!+a^{p-1}=p^k\] has only finitely many solutions?
Erdős and Graham remark that it is probably true that in general $(p-1)!+a^{p-1}$ is rarely a power at all (although this can happen, for example $6!+2^6=28^2$).