# Mersenne number

2010 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.

#### References

 [1] H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950) [2] A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)