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 for some positive integer . For example, is a Mersenne prime as it is a prime number and is expressible as . The exponents corresponding to Mersenne primes must themselves be prime, although the vast majority of primes do not lead to Mersenne primesâÂÂfor example, .
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, is a perfect number because the proper divisors of are , and , and .
Euclid proved that every prime expressed as has a corresponding perfect number . For example, the Mersenne prime leads to the corresponding perfect number . In 1747, Leonhard Euler completed what is now called the EuclidâÂÂEuler 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.
It is currently an open problem whether there are infinitely many Mersenne primes and even perfect numbers. The density of Mersenne primes is the subject of the LenstraâÂÂPomeranceâÂÂWagstaff conjecture, which states that the expected number of Mersenne primes less than some given is , where is Euler's number, is Euler's constant, and is the natural logarithm. It is widely believed, but not proven, that no odd perfect numbers exist; numerous restrictive conditions have been proven, including a lower bound of .
The following is a list of all 52 currently known () Mersenne primes and corresponding perfect numbers, along with their exponents . 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. New Mersenne primes are found using the LucasâÂÂLehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. Due to this efficiency, the largest known prime number has often been a Mersenne prime.
All possible exponents up to the 50th () have been tested and verified by GIMPS . Ranks 51 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.