# List of Mersenne primes and perfect numbers

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January 18, 2022

Mersenne primes and perfect numbers are two types of natural numbers that are closely related in number theory. Mersenne primes named after the mathematician Marin Mersenne are primes that can be expressed in the form 2p − 1 where p is a positive integer. For example, 3 is a Mersenne prime because it is prime and can be expressed as 22 − 1. The p numbers corresponding to Mersenne primes themselves must be prime, but not vice versa. every prime p leads to a Mersenne prime — for example, 211 − 1 2047 23 × 89. A perfect number is a natural number that is the sum of its own positive divisors, excluding the divisor of that very number. Accordingly, 6 is a perfect number because there are divisors (excluding 6) of 1, 2, 3 and 1 + 2 + 3 6. There is a duality between Mersenne primes and even perfect numbers. stated in the Euclid–Euler Theorem, partially proved by Euclid and perfected by Leonhard Euler: even numbers are perfect if and only if expressed as 2p − 1 × (2p − 1), where 2p − 1 is a Mersenne prime. In other words, any number that can be expressed in that form is a perfect number, and all even perfect numbers have that form. For example, when p 2, 22 − 1 3 is a prime number and 22 − 1 × (22 − 1) 2 × 3 6 is a perfect number. An open problem that does not currently have an answer is a number. Are Mersenne primes and even perfect numbers infinite? The frequency distribution of Mersenne primes is mentioned by the Lenstra – Pomerance – Wagstaff conjecture which states that the number of Mersenne primes less than x is given (eγ / log 2) × log log x, where e is the number Euler, γ is the Euler constant and log is the natural logarithm. It is not clear whether any odd perfect numbers exist; as well as various conditions for these numbers to be possible, such as if so their lower limit is 101500. The list below lists all Mersenne primes and now perfect numbers. known by the corresponding exponent p. As of 2021, 51 Mersenne primes have been discovered (corresponding to 51 perfect numbers), the 17 largest of which were discovered thanks to the Great Internet Mersenne Prime Search distributed computing project. The Internet Giant Mersenne) is abbreviated as GIMPS. New Mersenne primes were found using the Lucas-Lehmer Test (Lucas-Lehmer test - LLT), a primality test for Mersenne primes in a way that is efficient for binary computers. sorted in ascending order. As of 2021 there is still a small probability that the ranking may change if a smaller number is detected. According to GIMPS, all possibilities less than 48th appropriate exponent of p57,885.161 have been tested and verified through October 2021. Year and discoverer are timed for primes. Mersenne, because perfect numbers are calculated by a corollary of the Euclid-Euler theorem. "GIMPS/name" is used to refer to primes discovered by GIMPS and the individual who discovered them. The following numbers are too long to be written in the framework, so only the first 6 digits and the last 6 digits are displayed. List