Let $p$ be a prime number. Find the number of incongruent solutions of $$ x^{p^5}-x+p\equiv0\mod p^{2020}.$$
Let $f(x) = x^{p^5}-x+p$. Because of $f '(x)$ different from zero mod $p$. Then I say $$f(x) \equiv 0\mod p \quad(1) $$ and $$ f(x) \equiv0\mod p^k $$ are the same.
Then I calculate (1) equation. But in this part I'm not sure. For $x=0$ and $x=p$ it is true. But is there something else? How can I sure about it.
Let $f(x) = x^{p^5}-x+p$. Observe that $f(x) \equiv 0 \pmod p$ has exactly $p$ solutions. Moreover, $$f'(x) \equiv -1 \not\equiv 0 \pmod p,$$ for all $x \in \mathbb{Z}$. Thus, by Hensel's Lemma, each of these $p$ solutions can be lifted uniquely to a root modulo $p^2, p^3, \ldots, p^{2020}$. Accordingly, the congruence equation $f(x) \equiv 0 \pmod{p^{2020}}$ has $p$ unique incongruent solutions.