A Mersenne number is a number of the form
 |
(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, ... (Sloane's A000225), corresponding to 
,
,
,
,
The Mersenne numbers are also the numbers obtained by setting x = 1 in a Fermat polynomial.
The number of digits D in the Mersenne number 
is
 |
(2) |
where 
is the floor function, which, for large n, gives
 |
(3) |
The number of digits in is the same as the number of digits in 
,
The numbers of prime factors of for n = 1, 2, ... are 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, ... (Sloane's A046051), and the first few factorizations are
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(4) |
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|
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(5) |
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|
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(6) |
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|
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(7) |
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|
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(8) |
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|
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(9) |
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|
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(10) |
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|
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(11) |
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|
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(12) |
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|
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(13) |
(Sloane's A001265). The smallest primes dividing
are therefore 1, 3, 7, 3, 31, 3, 127, 3, 7, 3, 23, 3, 8191, ... (Sloane's A049479), and the largest are 1, 3, 7, 5, 31, 7, 127, 17, 73, 31, 89, 13, 8191, ... (Sloane's A005420).
In order for the Mersenne number to be prime, n must be prime. This is true since for composite n with factors r and s, n = rs. Therefore, 
can be written as 
,
 |
(14) |
but with p restricted to prime values.
All known Mersenne numbers 
with p prime are squarefree. However, Guy (1994) believes that there are 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. Eddington maintains a list of known factorizations of for prime p.
Catalan-Mersenne Number, Cunningham Number, Double Mersenne Number, Eberhart's Conjecture, Erdos-Borwein Constant, Fermat Number, Lucas-Lehmer Test, Mersenne Prime, Perfect Number, Repunit, Riesel Number, Sierpinski Number of the Second Kind, Sophie Germain Prime, Superperfect Number, Wheat and Chessboard Problem, Wieferich Prime, Zsigmondy Theorem
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 13, 1952.
Eddington, W. "Will Eddington'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 
[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.
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, A0054202609, A034887, A046051, and A049479 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 23-24, 1999.