Mersenne Prime

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a prime power of two; hence,

M_n

(most sources restrict the term Mersenne number to where ''n'' is prime as all Mersenne primes must be of this form as seen below)

Mersenne primes have a close connection to Perfect Number s, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all Even perfect numbers have this form. No Odd perfect numbers are known, and it is suspected that none exists (any that do have to belong to a significant number of special forms; see Perfect Number for more details).

It is currently unknown whether there is an Infinite Number Of Mersenne Primes .

SEARCHING FOR MERSENNE PRIMES

The identity

:

2^{ab}-1=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}
ight)

shows that ''M_n'' can be prime only if ''n'' itself is prime, which simplifies the search for Mersenne primes considerably. The converse statement, namely that ''M_n'' is necessarily prime if ''n'' is prime, is false. The smallest counterexample is 2¹¹ − 1 = 23 × 89, a Composite Number .

Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.

The first four Mersenne primes ''M''₂, ''M''₃, ''M''₅, ''M''₇ were known in antiquity.
The fifth, ''M''₁₃, was discovered anonymously before 1461; the next two (''M''₁₇ and ''M''₁₉) were found by Cataldi in 1588. After more than a century ''M''₃₁ was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was ''M''₁₂₇, found by Lucas in 1876, then ''M''₆₁ by Pervushin in 1883. Two more - ''M''₈₉ and ''M''₁₀₇ - were found early in the 20th century, by Powers in 1911 and 1914, respectively.

The numbers are named after 17th century French Mathematician Marin Mersenne , who provided a list of Mersenne primes with exponents up to 257. Unfortunately his list was not correct, as he mistakenly included ''M''₆₇ and ''M''₂₅₇, and omitted ''M''₆₁, ''M''₈₉ and ''M''₁₀₇.

The best method presently known for testing the primality of Mersenne numbers is based on the computation of a recurring Sequence , as developed originally by Lucas in 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer Test For Mersenne Numbers . Specifically, it can be shown that (for

n>2

)

M_n=2^n-1

is prime if and only if ''M_n'' divides ''S_n-2'', where

S_0=4

and for

k>0

S_k=S_{k-1}^2-2

.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. The first successful identification of a Mersenne prime, ''M''₅₂₁, by this means was achieved at 10:00 P.M. on January 30 , 1952 using the U.S. National Bureau Of Standards Western Automatic Computer (SWAC) at the Institute For Numerical Analysis at the University Of California, Los Angeles , under the direction of Lehmer , with a computer search program written and run by Prof. R.M. Robinson . It was the first Mersenne prime to be identified in thirty-eight years; the next one, ''M''₆₀₇, was found by the computer a little less than two hours later. Three more — ''M''₁₂₇₉, ''M''₂₂₀₃, ''M''₂₂₈₁ — were found by the same program in the next several months. ''M''₄₂₅₃ is the first Mersenne prime that is Titanic , and ''M''₄₄₄₉₇ is the first Gigantic .

As Of December 2005 , only 43 Mersenne primes are known; the largest known prime number (2^30,402,457 − 1) is a Mersenne prime. Like several previous Mersenne primes, it was discovered by a Distributed Computing project on the Internet, known as the '' Great Internet Mersenne Prime Search '' (GIMPS).

A log fit of the first 43 known prime exponents places the 44th exponent around 6E7, yielding a prime of 18 million digits. A lower predicted value falls out based on the Wagstaff conjecture.

THEOREMS ABOUT MERSENNE PRIME

If ''n'' is a positive integer,

:

c^n-d^n=(c-d)\sum_{k=0}^{n-1} c^kd^{n-1-k}

,

or

:

(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}
ight)=2^{ab}-1

by setting

c=2^a

d=1

, and

n=b

''proof''

:

(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k}

=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}

=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n

=a^n-b^n

2^n-1

is prime, then

n

is prime.

''proof''

By

:

(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}
ight)=2^{ab}-1

n

is not prime, or

n=ab

where

1 < a, b < n

.
Therefore,

2^a-1

would divide

2^n-1

,
or

2^n-1

is not prime.

LIST OF KNOWN MERSENNE PRIMES

The table below lists all known Mersenne primes :

It is not known whether any undiscovered Mersenne primes exist between the 38th (''M''_6972593) and the 43rd (''M''_30402457) on this chart; the ranking is therefore provisional.

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Mersenne Prime