Primes Powers and Mods The Question is below:
For which primes $p$ is $(p − 1)^p + 1$ a power of $p$?
I think the answer is $2$ and $3$, none of the others work.
Here is what I have:
let $p^k-1=(p-1)^p$. Then we have $$p^{k-1}+...+p+1 \equiv0 \pmod{p-1}.$$

Please also include a proof of your answer

If $(p-1)^p+1=p^k$ then
\begin{align*}
k&=\log_p \left(1+(p-1)^p\right)\\
&=\log_p \left(p^p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)\right)\\
&=p+\log_p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)
\end{align*}
clearly
$$\log_p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)<0$$
since $\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p<1$
and so it suffices to show that
$$\log_p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)>-1$$
for $p>3$ since then $p+\log_p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)$ cannot be an integer for $p>3$. Differentiating $\left(1-\frac{1}{x}\right)^x$ we get that it is an increasing function and so for $p\geq5$ we get that
$$\left(1-\frac{1}{x}\right)^x\geq\left(1-\frac{1}{5}\right)^5>.3$$
for $p\geq5$ we also clearly have that $\frac{1}{p}\leq \frac{1}{5}=.2$: it is thus that
\begin{align*}
\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p &> \left(1-\frac{1}{p}\right)^p\\
&>.3\\
&>.2\\
&\geq \frac{1}{p}
\end{align*}
taking logs base $p$ of both sides gets that
$$\log_p \left(\frac{1}{p^p}+\left(1-\frac{1}{p}\right)^p\right)>-1$$
and our proof is complete.
