List_of_Mersenne_primes_and_perfect_numbers

List of Mersenne primes and perfect numbers

List of Mersenne primes and perfect numbers

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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 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 numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89.[3] Meanwhile, 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]

Cuisenaire rods showing the proper divisors of 6 (1, 2, and 3) adding up to 6
Visualization of 6 as a perfect number
A graph plotting years on the x-axis with the number of digits of the largest known prime logarithmically on the y-axis, with two trendlines
Logarithmic graph of the number of digits of the largest known prime by year, nearly all of which have been Mersenne primes

There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. 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 2p − 1 × (2p − 1), where 2p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of p = 2, 22 − 1 = 3 is prime, and 22 − 1 × (22 − 1) = 2 × 3 = 6 is perfect.[1][5][6]

It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers.[2][6] 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 (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm.[7][8][9] 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 101500.[10]

The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. 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.[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]

The displayed ranks are among indices currently known as of 2022; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of January 2024.[11] The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last six digits of each number are shown.

More information Rank, p ...

Historically, the largest known prime number has often been a Mersenne prime.

Notes

  1. 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.
  2. 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.[16]
  3. Found in an anonymous manuscript, Clm 14908, dated 1456 and 1461, and in Ibn Fallus' earlier work, which was not widely distributed[14][17]
  4. 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.
  5. As of 27 April 2024[11]
  6. It has not been verified whether any undiscovered Mersenne primes exist between the 48th (M57,885,161) and the 51st (M82,589,933) on this table; the ranking is therefore provisional.
  7. 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.

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