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Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The ...
The smallest composite Mersenne number with prime exponent n is 2 11 − 1 = 2047 = 23 × 89. Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are ...
This is a list of articles about prime numbers. ... As of 2024, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and ...
All Mersenne primes are of the form M p = 2 p − 1, where p is a prime number itself. The smallest Mersenne prime in this table is 2 1398269 − 1. The first column is the rank of the Mersenne prime in the (ordered) sequence of all Mersenne primes; [33] GIMPS has found all known Mersenne primes beginning with the 35th. #
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
The historic finding is classified as a Mersenne prime, which is named after the French monk Marin Mersenne, who studied these numbers more than 350 years ago. Mersenne primes are a rare kind of ...
The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physico-Mathematica (1644; see e.g. Dickson 1919) that the numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n ≤ 257.
Base-2 repunit primes are called Mersenne primes. ... Therefore, the number cannot be prime. Base 20 repunit primes. The first few base-20 repunit primes are