When is $(p - 1)! + 1$ a power of $p$?

Solution 1:

Let $p^k - 1 = (p-1)!$. Cancel the factor of $(p-1)$, we get $$p^{k-1} + \cdots + 1 = (p-2)!$$ Now consider mod $p-1$ to get information about $k$. When can the equality ever hold?

Solution 2:

Not sure if only one hint would help, you could try using more hints as needed. First, the statement holds for all natural numbers $n \notin \{2, 3, 5\} $, not just for primes. One way to proceed is to show that for $n > 5$:
1) $n$ cannot be even if it satisfies the equation
2) for $n$ odd, $(n-1)^2 \space | \space (n-1)!$
3) from (2) above and the given equation, show $(n-1) \space | \space k$ which implies $n-1 \le k$
4) show that $\forall n > 2, (n-1)! < n^{n-1} - 1$

Hope that helps!