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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 ...
Since Mersenne numbers grow very rapidly, the search for Mersenne primes is a difficult task, even though there is a simple efficient test to determine whether a given Mersenne number is prime: the Lucas–Lehmer primality test (LLT), which makes it much easier to test the primality of Mersenne numbers than that of most other numbers of the ...
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. #
The largest known prime number is 2 136,279,841 − 1, a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).
The Lucas–Lehmer test is one of the main primality tests used by the Great Internet Mersenne Prime Search (GIMPS) to locate large primes. This search has been successful in locating many of the largest primes known to date. [10] The test is considered valuable because it can provably test a large set of very large numbers for primality within ...
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 (sequence A109461 in the OEIS), and were composite for all other positive integers n ≤ 257.
As of 2024, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. This includes the largest known prime 2 136,279,841-1, which is the 52nd Mersenne prime.
Search. Search. Appearance. Donate; ... Base-2 repunit primes are called Mersenne primes. ... Therefore, the number cannot be prime. Base 5 repunit primes