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 M_n = 2 − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2 − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form M_p = 2 − 1 for some prime p.

The exponents n that give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).

Numbers of the form M_n = 2 − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n should be prime. The smallest composite Mersenne number with prime exponent n is 2 − 1 = 2047 = 23 × 89.

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid-Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

As of 2025, 52 Mersenne primes are known. The largest known prime number, 2 − 1, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.

A basic theorem about Mersenne numbers states that if M_p is prime, then the exponent p must also be prime. This follows from the identity

\(\begin{aligned} 2^{ab}-1 &=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)\\ &=(2^b-1)\cdot \left(1+2^b+2^{2b}+2^{3b}+\cdots+2^{(a-1)b}\right). \end{aligned}\)

This rules out primality for Mersenne numbers with a composite exponent, such as M₄ = 2 − 1 = 15 = 3 × 5 = (2 − 1) × (1 + 2).

Though the above examples might suggest that M_p is prime for all primes p, this is not the case, and the smallest counterexample is the Mersenne number

M₁₁ = 2 − 1 = 2047 = 23 × 89.

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite.

The Lenstra-Pomerance-Wagstaff conjecture claims that there are infinitely many Mersenne primes and predicts their order of growth and frequency: For every number n, there should on average be about \(e^\gamma\cdot\log_2(10) \approx 5.92\) primes p with n decimal digits for which M_p is prime. Here, γ is the Euler-Mascheroni constant.

It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers; for example, from the infinitude of Sophie Germain primes congruent to 3 (mod 4). For these primes p, 2p + 1 (which is also prime) will divide M_p, for example, 23 | M₁₁, 47 | M₂₃, 167 | M₈₃, 263 | M₁₃₁, 359 | M₁₇₉, 383 | M₁₉₁, 479 | M₂₃₉, and 503 | M₂₅₁ (sequence A002515 in the OEIS). For these primes p, 2p + 1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide \(\frac{(2p+1)-1}{2} = p\). Since p is a prime, it must be p or 1. However, it cannot be 1 since Φ₁(2) = 1 and 1 has no prime factors, so it must be p. Hence, 2p + 1 divides Φ_p(2) = 2 − 1 and 2 − 1 = M_p cannot be prime. The first four Mersenne primes are M₂ = 3, M₃ = 7, M₅ = 31 and M₇ = 127 and because the first Mersenne prime starts at M₂, all Mersenne primes are congruent to 3 (mod 4). Other than M₀ = 0 and M₁ = 1, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number ( ≥ M₂) there must be at least one prime factor congruent to 3 (mod 4).

The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime values of M_p appear to grow increasingly sparse as p increases. For example, eight of the first 11 primes p give rise to a Mersenne prime M_p (the correct terms on Mersenne's original list), while M_p is prime for only 43 of the first two million prime numbers (up to 32452843).

Condensed: the full section is in Wikipedia.

Mersenne primes M_p are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if 2 − 1 is prime, then 2(2 − 1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. This is known as the Euclid-Euler theorem. It is unknown whether there are any odd perfect numbers.

Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne in 1644 were as follows:

2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.

His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included M₆₇ and M₂₅₇ (which are composite) and omitted M₆₁, M₈₉, and M₁₀₇ (which are prime). Mersenne gave little indication of how he came up with his list.

Édouard Lucas proved in 1876 that M₁₂₇ is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Aimé Ferrier found a larger prime, (2 + 1)/17, using a desk calculating machine. M₆₁ was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876 by demonstrating that M₆₇ was composite without finding a factor. No factor was found until a famous talk by Frank Nelson Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one, resulting in the number 147573952589676412927. On the other side of the board, he multiplied 193707721 × 761838257287 and got the same number, then returned to his seat (to applause) without speaking. He later said that the result had taken him "three years of Sundays" to find. A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.

Fast algorithms for finding Mersenne primes are available, and as of October 2024, the seven largest known prime numbers are Mersenne primes.

The first four Mersenne primes M₂ = 3, M₃ = 7, M₅ = 31 and M₇ = 127 were known in antiquity. The fifth, M₁₃ = 8191, was discovered anonymously before 1461; the next two (M₁₇ and M₁₉) were found by Pietro Cataldi in 1588. After nearly two centuries, M₃₁ was verified to be prime by Leonhard Euler in 1772. The next (in historical, not numerical order) was M₁₂₇, found by Édouard Lucas in 1876, then M₆₁ by Ivan Mikheevich Pervushin in 1883. Two more (M₈₉ and M₁₀₇) were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.

The most efficient method presently known for testing the primality of Mersenne numbers is the Lucas-Lehmer primality test. Specifically, it can be shown that for prime p > 2, M_p = 2 − 1 is prime if and only if M_p divides S_{p − 2}, where S₀ = 4 and S_k = (S_{k − 1}) − 2 for k > 0.

During the era of manual calculation, all previously untested exponents up to and including 257 were tested with the Lucas-Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229. Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times as large as the previous record of 127.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949, but the first successful identification of a Mersenne prime, M₅₂₁, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles (UCLA), under the direction of D. H. Lehmer, with a computer search program written and run by Prof. R. M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M₆₀₇, was found by the computer a little less than two hours later. Three more – M₁₂₇₉, M₂₂₀₃, and M₂₂₈₁ – were found by the same program in the next several months. M₄₄₂₃ was the first prime discovered with more than 1000 digits, M₄₄₄₉₇ was the first with more than 10000, and M_6972593 was the first with more than a million. In general, the number of digits in the decimal representation of M_n equals ⌊n log₁₀2⌋ + 1, where ⌊x⌋ denotes the floor function (or equivalently ⌊log₁₀M_n⌋ + 1).

In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search (GIMPS) won part of a $100000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.

Condensed: the full section is in Wikipedia.

As of 2024, the 52 known Mersenne primes are 2 − 1 for the following p:

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933, 136279841. (sequence A000043 in the OEIS)

Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all Mersenne numbers are Mersenne primes. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019, 2 − 1 is the record-holder, having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See Integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then running a primality test on the cofactor. As of September 2022, the largest completely factored number (with probable prime factors allowed) is 2 − 1 = 1119429257 × 175573124547437977 × 8480999878421106991 × q, where q is a 3829294-digit probable prime. It was discovered by a GIMPS participant with nickname "Funky Waddle". As of September 2022, the Mersenne number M₁₂₇₇ is the smallest composite Mersenne number with no known factors; it has no prime factors below 2, and is very unlikely to have any factors below 10 (~2).

The table below shows factorizations for the first 20 composite Mersenne numbers where the exponent p is a prime number (sequence A244453 in the OEIS).

The number of factors for the first 500 Mersenne numbers can be found at (sequence A046800 in the OEIS).

In the mathematical problem Tower of Hanoi, optimally solving a puzzle with an n-disc tower requires M_n steps. The number of rice grains on the whole chessboard in the wheat and chessboard problem is M₆₄.

The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime.

In geometry, an integer right triangle that is primitive and has its even leg a power of 2 ( ≥ 4) generates a unique right triangle such that its inradius is always a Mersenne number. For example, if the even leg is 2 then because it is primitive it constrains the odd leg to be 4 − 1, the hypotenuse to be 4 + 1 and its inradius to be 2 − 1.

Also in geometry, the number of polytopes that are part of the family of polytopes formed by a truncation operation of a base regular polytope and its dual (excluding alternation) is equal to M_n where n is the dimension of the base polytope. For example, a tesseract and its dual the hexadecachoron have M₄ = 15 different polytopes in its family formed by truncation operations.

A Mersenne-Fermat number is defined as ⁠2 − 1/2 − 1⁠ with p prime, r natural number, and can be written as MF(p, r). When r = 1, it is a Mersenne number. When p = 2, it is a Fermat number. The only known Mersenne-Fermat primes with r > 1 are

MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2), and MF(59, 2).

In fact, MF(p, r) = Φ_p(2), where Φ is the cyclotomic polynomial.

The simplest generalized Mersenne primes are prime numbers of the form f(2), where f(x) is a low-degree polynomial with small integer coefficients. An example is 2 − 2 + 1, in this case, n = 32, and f(x) = x − x + 1; another example is 2 − 2 − 1, in this case, n = 64, and f(x) = x − x − 1.

It is also natural to try to generalize primes of the form 2 − 1 to primes of the form b − 1 (for b ≠ 2 and n > 1). However (see also § Theorems about Mersenne numbers above), b − 1 is always divisible by b − 1, so unless the latter is a unit, the former is not a prime. This can be remedied by allowing b to be an algebraic integer instead of an integer:

In the ring of integers (on real numbers), if b − 1 is a unit, then b is either 2 or 0. But 2 − 1 are the usual Mersenne primes, and the formula 0 − 1 does not lead to anything interesting (since it is always −1 for all n > 0). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.