Lucas-Lehmer Test: for *p* an odd prime, the Mersenne number 2^{p} − 1 is prime if and only if 2^{p} − 1 divides *S*(*p* − 1) where *S*(*n* + 1) = (*S*(*n*))^{2} − 2, and *S*(1) = 4.

The following programs calculate all Mersenne primes up to the implementation’s maximum precision, or the 47th Mersenne prime. (Which ever comes first).

#include <iostream> #include <gmpxx.h> static bool is_mersenne_prime(mpz_class p) { if( 2 == p ) return true; else { mpz_class s(4); mpz_class div( (mpz_class(1) << p.get_ui()) - 1 ); for( mpz_class i(3); i <= p; ++i ) { s = (s * s - mpz_class(2)) % div ; } return ( s == mpz_class(0) ); } } int main() { mpz_class maxcount(45); mpz_class found(0); mpz_class check(0); for( mpz_nextprime(check.get_mpz_t(), check.get_mpz_t()); found < maxcount; mpz_nextprime(check.get_mpz_t(), check.get_mpz_t())) { //std::cout << "P" << check << " " << std::flush; if( is_mersenne_prime(check) ) { ++found; std::cout << "M" << check << " " << std::flush; } } }

- Output:

(Incomplete; It takes a long time.)

M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689 M9941 M11213 M19937 M21701 M23209 M44497

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