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)