RO-NMO 2005 Problem 124

Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let $$M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} .$$ Prove that the following statements are equivalent:

a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$ .

Marian Andronache