The Lucas–Lehmer test

Test. For $p$ prime, $M_p=2^p-1$ is prime iff $s_{p-2}\equiv0\pmod{M_p}$, where $s_0=4$ and $s_{n+1}=s_n^2-2$.

Apply to $p=3$. $M_3=7$. $s_0=4$. $s_1=4^2-2=14\equiv0\pmod7$. Since $p-2=1$ and $s_1\equiv0$, $M_3$ is prime. ✓

Conclusion. Lucas–Lehmer is a fast test for Mersenne primes; used to find the largest known primes.