Lucas-Lehmer residue

The Lucas-Lehmer test is an efficient deterministic primality test for determining if a Mersenne number M_n is prime. Since it is known that Mersenne numbers can only be prime for prime subscripts, attention can be restricted to Mersenne numbers of the form M_p = 2^p - 1, where p is an odd prime. Consider the recurrence equation s_n congruent s_(n - 1)^2 - 2 (mod M_p) with s_0 congruent 4. For example, ignoring the congruence, the first few terms of this iteration are 4, 14, 194, 37634, 1416317954, ... (OEIS A003010).

Lucas sequence | Mersenne number | Mersenne prime | Pratt certificate | Rabin-Miller strong pseudoprime test