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 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1.[1][2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89.[3]
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.[2][4]
Euclid proved c.300 BCE that every prime expressed as Mp = 2p − 1 has a corresponding perfect number Mp × (Mp+1)/2 =2p − 1 × (2p − 1). For example, the Mersenne prime 22 − 1 = 3 leads to the corresponding perfect number 22 − 1 × (22 − 1) =2 × 3 = 6. In 1747, Leonhard Euler completed what is now called the Euclid–Euler theorem, showing that these are the only even perfect numbers. As a result, there is a one-to-one correspondence between Mersenne primes and even perfect numbers, so a list of one can be converted into a list of the other.[1][5][6]
The following is a list of all 52 currently known (as of January2025[update]) Mersenne primes and corresponding perfect numbers, along with their exponents p. The largest 18 of these have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS; their discoverers are listed as "GIMPS / name", where the name is the person who supplied the computer that made the discovery.[2] New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.[2] Due to this efficiency, the largest known prime number has often been a Mersenne prime.[12]
All possible exponents up to the 48th (p = 57,885,161) have been tested and verified by GIMPS as of October2021[update].[13] Ranks 49 and up are provisional, and may change in the unlikely event that additional primes are discovered between the currently listed ones. Later entries are extremely long, so only the first and last six digits of each number are shown, along with the number of decimal digits.
More information Rank, p ...
Table of all 52 currently-known Mersenne primes and corresponding perfect numbers
The first four perfect numbers were documented by Nicomachus circa 100, and the concept was known (along with corresponding Mersenne primes) to Euclid at the time of his Elements. There is no record of discovery.
Islamic mathematicians such as Ismail ibn Ibrahim ibn Fallus (1194–1239) may have known of the fifth through seventh perfect numbers prior to European records.[18]
Found in an anonymous manuscript designated Clm 14908, dated 1456 and 1461. Ibn Fallus' earlier work in the 13th century also mentioned the prime, but was not widely distributed.[16][19]
M42,643,801 was first reported to GIMPS on April 12, 2009, but was not noticed by a human until June 4, 2009, due to a server error.
As of 14May2025[update].[13] All exponents below the lowest unverified milestone have been checked more than once. All exponents below the lowest untested milestone have been checked at least once.
It has not been verified whether any undiscovered Mersenne primes exist between the 48th (M57,885,161) and the 52nd (M136,279,841) on this table; the ranking is therefore provisional.
M74,207,281 was first reported to GIMPS on September 17, 2015 but was not noticed by a human until January 7, 2016 due to a server error.
