Theory
M_p = 2^p − 1 (Mersenne number)
Necessary condition: p must be prime
(if p=ab, then 2^a−1 | 2^p−1)
Lucas-Lehmer Test:
s_0 = 4, s_{n+1} = s_n² − 2 (mod M_p)
M_p prime ⟺ s_{p-2} ≡ 0 (mod M_p)
Known: 51 Mersenne primes (2024)
Largest: 2^136,279,841 − 1 (GIMPS 2024)