MathNet 2024-part-04 — 168 problems (EN–VI) Question 153

Let $p \gt 2$ be a prime number and $n \in \mathbb{N}^*$ such that $n \equiv 1 \pmod{p}$. If the ring $(A, +, \cdot)$ has the property that $x^{n+p} = x^n$ for any $x \in A$, prove that $x^{p+1} = x$ for any $x \in A$.

Mihai Opincariu