The displayed ranks are among indices currently known as of 2022 while unlikely, ranks may change if smaller ones are discovered. New Mersenne primes are found using the Lucas-Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. As of 2023, there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. It is also not known if any odd perfect numbers exist various conditions on possible odd perfect numbers have been proven, including a lower bound of 10 1500. The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is ( e γ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. For instance, in the case of p = 2, 2 2 − 1 = 3 is prime, and 2 2 − 1 × (2 2 − 1) = 2 × 3 = 6 is perfect. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form.
This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p − 1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers.
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself.
The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes-for example, 2 11 − 1 = 2047 = 23 × 89. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Visualization of 6 as a perfect number Logarithmic graph of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes
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