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 $6!+2^6=28^2$).

Erdős and Graham ask this allowing the case $p=2$, but this is presumably an oversight, since clearly there are infinitely many solutions to this equation when $p=2$.

Brindza and Erdős [BrEr91] proved that there are finitely many such solutions. Yu and Liu [YuLi96] showed that the only solutions are\[2!+1^2=3\]\[2!+5^2=3^3\]and\[4!+1^4=5^2.\]

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