Theorem: Let $p>2$ be a prime, and let $M_p$ be a Mersenne number, then $M_p$ is prime iff $M_p$ divides $s_{p-1}$ where the numbers $(s_n)_{n\geq1}$ are given by the following recurrence relation: $s_1=4$ , and $$s_{n+1}={s_n}^2-2$$ for $n\geq1$ .