A Mersenne number is a number of the form
 |
(1)
|
where 
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 
,
, 
, 
, ... in binary.
The Mersenne numbers are also the numbers obtained by setting 
in a Fermat polynomial.
They also correspond to Cunningham numbers 
.
The number of digits 
in the Mersenne number 
is
 |
(2)
|
where 
is the floor function, which, for large 
, gives
 |
(3)
|
The number of digits in 
is the same as the number of digits in 
, 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 
for 
,
1, ... are given by 1, 4, 31, 302, 3011, 30103, 301030, 3010300, 30103000, 301029996,
... (OEIS A114475), which correspond to the
decimal expansion of 
(OEIS A007524).
The numbers of prime factors of 
for 
, 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
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(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 
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 
to be prime, 
must be prime. This is true
since for composite 
with factors 
and 
, 
. Therefore, 
can be written as 
, 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
 |
(14)
|
but with 
restricted to prime values.
All known Mersenne numbers 
with 
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. Edgington maintains a list of known factorizations of 
for prime 
.
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