Are there an infinite amount of Mersenne primes?
Are there an infinite amount of Mersenne primes?
As of October 2020, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime….Mersenne prime.
| Named after | Marin Mersenne |
|---|---|
| No. of known terms | 51 |
| Conjectured no. of terms | Infinite |
| Subsequence of | Mersenne numbers |
| First terms | 3, 7, 31, 127, 8191 |
What is the largest known Mersenne prime?
2^82,589,933 – 1
The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933 – 1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida, made the find on December 7, 2018.
Are there infinite prime numbers?
The number of primes is infinite. The first ones are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 and so on. The first proof of this important theorem was provided by the ancient Greek mathematician Euclid. His proof is known as Euclid’s theorem.
Is 2047 a Mersenne prime?
A Mersenne prime is a Mersenne number that is a prime number. For example, 31 = 25 − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 211 − 1 is not a prime because it is divisible by 89 and 23.
Is 13 a Mersenne prime?
, 3, 5, 7, 13, 17, 19, 31, 61, 89, (OEIS A000043).
Are there any other Mersenne numbers that are prime?
Other than M0 = 0 and M1 = 1, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number ( ≥ M2 ) there must be at least one prime factor congruent to 3 (mod 4). A basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime.
Which is an example of a generalized Mersenne prime?
The simplest generalized Mersenne primes are prime numbers of the form f(2n), where f(x) is a low-degree polynomial with small integer coefficients. An example is 264 − 232 + 1, in this case, n = 32, and f(x) = x2 − x + 1; another example is 2192 − 264 − 1, in this case, n = 64, and f(x) = x3 − x − 1 .
Which is the best test for the primality of Mersenne numbers?
The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, M p = 2 p − 1 is prime if and only if M p divides S p − 2, where S 0 = 4 and S k = (S k − 1) 2 − 2 for k > 0.
Which is the smallest Mersenne number with prime exponent n?
More generally, numbers of the form M n = 2 n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 2 11 − 1 = 2047 = 23 × 89.
