A Mersenne prime is a prime of the form \(M_p = 2^p - 1\), where \(p\) itself must also be prime (though not every prime \(p\) produces a Mersenne prime — \(M_{11} = 2047 = 23 \times 89\) is composite).
The known Mersenne primes start \(M_2=3\), \(M_3=7\), \(M_5=31\), \(M_7=127\), \(M_{13}=8191\). They are named after the 17th-century French monk Marin Mersenne who studied them.
Mersenne primes are the largest known primes by far, due to the efficient Lucas–Lehmer primality test that works only for them. The Great Internet Mersenne Prime Search (GIMPS) is a long-running distributed computing project that has discovered all the largest known primes since 1996.