Mersenne number

A prime number of the form , where . Mersenne numbers were considered in the 17th century by M. Mersenne. The numbers can be prime only for prime values of . For one obtains the prime numbers . However, for the number is composite. For prime values of larger than , among the one encounters both prime and composite numbers. The fast growth of the numbers makes their study difficult. By considering concrete numbers it has been shown, for example, that (L. Euler, 1750) and (I.M. Pervushin, 1883) are Mersenne numbers. Computers were used to find other very large Mersenne numbers, among them . 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)

Comments

Presently (1989) it is known that for the following the Mersenne number is prime: 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, 132049, 216091. See [a1].

The Lucas test provides a very simple method to establish primality of these numbers. This test consists of the following (cf. [a2]). Define and for . Then is prime if and only if divides (and is a prime number).

References

[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)