4. Let $p$ be a prime. Find all positive integer solutions to the indeterminate equation $p^{x}-y^{p}=1$.

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If $y>1$, and $p \neq 3$, then by Sigmund's theorem, we know that $y^{p}+1$ has a prime factor that does not divide $y+1$. But $(y+1) \mid (y^{p}+1)$, which implies that $y^{p}+1$ has at least two prime factors, a contradiction.
$$
\text { Hence }(p, x, y)=(2,1,1),(3,2,2) \text {. }
$$