A Mersenne prime is in the form of \(2^S-1\). Known Mersenne prime numbers are \(2^{3}-1\), \(2^{5}-1\), \(2^{7}-1\), \(2^{13}-1\), \(2^{17}-1\), \(2^{19}-1\), \(2^{31}-1\), \(2^{61}-1\), \(2^{89}-1\), \(2^{107}-1\) and \(2^{127}-1\). Overall Mersenne primes are efficient in their implementation.
Mersenne numbers |
Theory
import galois import sys n=2 if (len(sys.argv)>1): n=int(sys.argv[1]) print("Mersenne_exponents:", galois.mersenne_exponents(n)) print("Mersenne primes:", galois.mersenne_primes(n))
and a sample run:
Mersenne_exponents: [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607] Mersenne primes: [3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151, 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127]