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A *Mersenne Prime* is a prime number that is one less than a power of two.
That is, a number of the form: 2^{n} - 1, for some number “n”.
This can also be written as 2^n - 1,
for browsers that do not display subscripts and superscripts.

Mersenne primes are named after Marin Mersenne (1588-1648), a French monk and mathematician.

It is easier (i.e., it takes less computation) to determine whether a Mersenne number is prime, than to test other numbers of the same size. For this reason, the largest known prime is usually a Mersenne prime.

The first few Mersenne primes are:

2 ^{2} - 1 = 2 * 2 - 1 = 3

2 ^{3} - 1 = 2 * 2 * 2 - 1 = 7

2 ^{5} - 1 = 2 * 2 * 2 * 2 * 2 - 1 = 31

2 ^{7} - 1 = 2 * 2 * 2 * 2 * 2 * 2 * 2 - 1 = 127

The next six Mersenne primes correspond to the following values for the exponent “n”: 13, 17, 19, 31, 61, and 89.

In order for 2^{n} - 1 to be prime,
the exponent “n” must also be prime.
However, if “n” is prime,
this does not guarantee that 2^{n} - 1 is prime.
For example, 11 is a prime number,
but 2^{11} - 1 = 2047 is **not** prime
(and therefore not a Mersenne prime),
because 2047 = 23 * 89.

A list of all known Mersenne primes (only 49 of them) is available here.

Mersenne is only one of many mathematicians
with numbers named after him.
Pierre de Fermat (1601-1665),
who is famous for his “Last Theorem”,
also has some numbers named after him.
A Fermat number is a number of the form 2^{(2^n)} + 1.
Or 2^(2^n) + 1.
Fermat knew that the first five numbers (for “n” = 0 to 4)
of this form are prime,
and he believed that ALL numbers of this form are prime.
However, these five Fermat numbers are still the only ones
that are known to be prime,
and many are known to be composite.

n 2^n 2^(2^n) + 1 -------------------------------------- 0 1 2 + 1 = 3 = PRIME 1 2 4 + 1 = 5 = PRIME 2 4 16 + 1 = 17 = PRIME 3 8 256 + 1 = 257 = PRIME 4 16 65536 + 1 = 65537 = PRIME 5 32 2^32 + 1 = 4294967297 = 641 * 6700417 6 64 2^64 + 1 = 18446744073709551617 = 274177 * 67280421310721The Fermat numbers F

In 1999, Fermat number F_{24}, which is 2^{16777216} + 1,
and has more than 5,000,000 digits,
was proven to be composite
using
Pepin’s test.

Until recently, F_{31} with over 600,000,000 digits,
was the smallest Fermat number
whose character (prime or composite) was unknown.
On April 12, 2001, Alexander Kruppa discovered that
46931635677864055013377 = 5463561471303 * 2^33 + 1
is a factor of F_{31}.
Now the smallest Fermat number of unknown character is F_{33},
which has 2,585,827,974 digits!

Many Fermat numbers larger than F_{33} are known to be composite,
but only because someone found a prime number
that happened to divide evenly into one of these numbers.
The current status of the factoring of Fermat numbers is
available here.

E-mail me at jrhowell@ix.netcom.com

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This page was last updated on April 6, 2017.