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