# Mersenne number

2020 Mathematics Subject Classification: *Primary:* 11A [MSN][ZBL]

*Mersenne prime*

A prime number of the form $M_n=2^n-1$, where $n=1,2,\ldots$. Mersenne numbers were considered in the 17th century by M. Mersenne. The numbers $M_n$ can be prime only for prime values of $n$, since if $d$ divides $n$ then $M_d$ divides $M_n$. For $n=2,3,5,7$ one obtains the prime numbers $M_n=3,7,31,127$. However, for $n=11$ the number $M_n$ is composite. For prime values of $n$ larger than $11$, among the $M_n$ one encounters both prime and composite numbers. The fast growth of the numbers $M_n$ makes their study difficult. By considering concrete numbers $M_n$ it has been shown, for example, that $M_{31}$ (L. Euler, 1750) and $M_{61}$ (I.M. Pervushin, 1883) are Mersenne numbers. Computers were used to find other very large Mersenne numbers, among them $M_{11213}$. The existence of an infinite set of Mersenne numbers is still an open problem (1989). This problem is closely related with the problem on the existence of perfect numbers.

#### Comments

The exponents $n$ such that the Mersenne number $M_n$ is prime that were known before 1970 are

$$2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213.$$

More have been found before 1989 : $$19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091.$$

In 2023, 51 such exponents are known, the largest one being $82589933$ found in December 2018 by the Great Internet Mersenne Prime Search.

The Lucas test provides a very simple method to establish primality of these numbers. This test consists of the following (cf. [a2]). Define $S_1=4$ and $S_{k+1}=S_k^2-2$ for $k\geq1$. Then $M_n$ is prime if and only if $M_n$ divides $S_{n-1}$ (and $n$ is a prime number).

#### References

[1] | H. Hasse, "Vorlesungen über Zahlentheorie", Springer (1950) |

[2] | A.A. Bukhshtab, "Number theory", Moscow (1966) (In Russian) |

[a1] | H. Riesel, "Prime numbers and computer methods for factorisation", Birkhäuser (1986) |

[a2] | D. Shanks, "Solved and unsolved problems in number theory", Chelsea, reprint (1978) |

**How to Cite This Entry:**

Mersenne number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=54793