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Mersenne Number


A Mersenne number is a number of the form

 M_n=2^n-1,
(1)

where n is an integer. The Mersenne numbers consist of all 1s in base-2, and are therefore binary repunits. The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, ... (OEIS A000225), corresponding to 1_2, 11_2, 111_2, 1111_2, ... in binary.

The Mersenne numbers are also the numbers obtained by setting x=1 in a Fermat polynomial. They also correspond to Cunningham numbers C^-(2,n).

The number of digits D in the Mersenne number M_n is

 D=|_log(2^n-1)+1_|,
(2)

where |_x_| is the floor function, which, for large n, gives

 D approx |_nlog2+1_| approx |_0.301029n+1_|=|_0.301029n_|+1.
(3)

The number of digits in M_n is the same as the number of digits in 2^n, namely 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, ... (OEIS A034887). The numbers of decimal digits in M_(10^n) for n=0, 1, ... are given by 1, 4, 31, 302, 3011, 30103, 301030, 3010300, 30103000, 301029996, ... (OEIS A114475), which correspond to the decimal expansion of log2=0.30102999... (OEIS A007524).

The numbers of prime factors of M_n for n=1, 2, ... are 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, ... (OEIS A046051), and the first few factorizations are

M_1=1
(4)
M_2=3
(5)
M_3=7
(6)
M_4=3·5
(7)
M_5=31
(8)
M_6=3·3·7
(9)
M_7=127
(10)
M_8=3·5·17
(11)
M_9=7·73
(12)
M_(10)=3·11·31
(13)

(OEIS A001265). The indices of Mersenne numbers that give semiprimes are 4, 9, 11, 23, 37, 41, 49, 59, 67, 83, ... (OEIS A085724). As of 2022, the largest known indices giving semiprimes are 1427 and 1487.

The smallest primes dividing M_n are therefore 1, 3, 7, 3, 31, 3, 127, 3, 7, 3, 23, 3, 8191, ... (OEIS A049479), and the largest are 1, 3, 7, 5, 31, 7, 127, 17, 73, 31, 89, 13, 8191, ... (OEIS A005420).

In order for the Mersenne number M_n to be prime, n must be prime. This is true since for composite n with factors r and s, n=rs. Therefore, 2^n-1 can be written as 2^(rs)-1, which is a binomial number and can be factored. Since the most interest in Mersenne numbers arises from attempts to factor them, many authors prefer to define a Mersenne number as a number of the above form

 M_p=2^p-1,
(14)

but with p restricted to prime values.

All known Mersenne numbers M_p with p prime are squarefree. However, Guy (1994) believes that there are M_p which are not squarefree.

The search for Mersenne primes is one of the most computationally intensive and actively pursued areas of advanced and distributed computing. Edgington maintains a list of known factorizations of M_p for prime p.


See also

Catalan-Mersenne Number, Cullen Number, Cunningham Number, Double Mersenne Number, Eberhart's Conjecture, Erdős-Borwein Constant, Fermat Number, Lucas-Lehmer Test, Mersenne Prime, Perfect Number, Repunit, Riesel Number, Sierpiński Number of the Second Kind, Sophie Germain Prime, Superperfect Number, Wheat and Chessboard Problem, Wieferich Prime, Woodall Number, Zsigmondy Theorem

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References

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 13, 2005.Edgington, W. "Will Edgington's Mersenne Page." http://www.garlic.com/~wedgingt/mersenne.html.Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 47-51, 2000.Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150-156, May 1957.Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape k·2^n+2 [sic]." §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 15-16 and 22, 1979.Pappas, T. "Mersenne's Number." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 211, 1989.Robinson, R. M. "Mersenne and Fermat Numbers." Proc. Amer. Math. Soc. 5, 842-846, 1954.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 14, 18-19, 22, and 29-30, 1993.Sloane, N. J. A. Sequences A000225/M2655, A001265, A005420/M2609, A007524/M2196, A034887, A046051, A049479, A085724, and A114475 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 23-24, 1999.

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Mersenne Number

Cite this as:

Weisstein, Eric W. "Mersenne Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MersenneNumber.html

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