Digits in largest prime found as a function of time

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...

s and

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

s are two deeply interlinked types of

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

. Mersenne primes, named after the friar

Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...

, are

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s 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 numbers corresponding to Mersenne primes must themselves be prime, although not all primes lead to Mersenne primes—for example, . Meanwhile, perfect numbers are

natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

that equal the sum of their positive

proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s, which are divisors excluding the number itself. So, is a perfect number because the proper divisors of are , and , and . There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the

Euclid–Euler theorem The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...

, partially proved by

Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

and completed by

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

: even numbers are perfect

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

they can be expressed in the form , where 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 , is prime, and is perfect. It is currently an

open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...

as to whether there are an infinite number of Mersenne primes and even perfect numbers. The frequency of Mersenne primes is the subject of the

Lenstra–Pomerance–Wagstaff conjecture In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one. Original Mersenne conjecture The original, called Mersenne's conjectur ...

, which states that the expected number of Mersenne primes less than some given is , where is Euler's number, is

Euler's constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...

, and is the

natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...

. 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 In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

of . The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents . , there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the

distributed computing A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed com ...

project

Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...

, or GIMPS. New Mersenne primes are found using the Lucas-Lehmer test (LLT), a

primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...

for Mersenne primes that is efficient for binary computers. The displayed ranks are among indices currently known ; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent have been checked and verified . 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 6 digits of each number are shown. Historically, the

largest known prime number The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a posi ...

has often been a Mersenne prime.

Notes

References

External links

* * *
List on GIMPS, with the full values of large numbers

A technical report on the history of Mersenne numbers, by Guy Haworth
{{DEFAULTSORT:Mersenne primes and perfect numbers Mathematical tables Mathematics-related lists List List