Definition:Mersenne Prime

A Mersenne prime is a Mersenne number which happens to be prime.

That is, it is a prime number of the form $$2^p - 1$$.

From Primes of the Form of a Power Less One, it is clear that in order for $$2^p - 1$$ to be prime, then $$p$$ must also be prime.

The number $$2^p - 1$$ is, in this context, often denoted $$M_p$$.

It is not known whether there is an infinite number of Mersenne primes.

Historical Note
They are named for Marin Mersenne, who was the first on record to investigate them.

Mersenne published a book "Cogitata Physico-Mathematica" in 1644, in which he claimed that the only primes $$p \le 257$$ for which $$2^p - 1$$ is prime are $$2, 3, 5, 7, 13, 17, 19, 31, 67, 127$$ and $$257$$.

He was not entirely correct.


 * 1732: Euler showed that $$M_{29}$$ is composite


 * 1772: Euler showed that $$M_{31}$$ is indeed prime.


 * 1811: Peter Barlow (somewhat short-sightedly, given historical 20-20 hindsight) stated in his book "Theory of Numbers" that "$$M_{31}$$ is the greatest prime that will ever be found."


 * 1876: Édouard Lucas proved that $$M_{127}$$ is prime, and also discovered that $$M_{67}$$ is actually composite.


 * 1883: Ivan Pervushin proved that $$M_{61}$$ is prime.


 * 1903: The factors of were found by Frank Cole who delivered a now famous lecture "On The Factorization of Large Numbers" in which he performed (without uttering a word) the arithmetic demonstrating what those factors were.


 * 1911: R. E. Powers proved that $$M_{89}$$ is prime.


 * 1914: R. E. Powers proved that $$M_{107}$$ is prime.


 * 1922: Maurice Kraitchik proved that $$M_{257}$$ is actually composite.

Thus Mersenne's assertion was finally investigated in full: he had been determined to be wrong by including $$M_{67}$$ and $$M_{257}$$ in his list of primes, and by failing to include $$M_{61}$$, $$M_{89}$$ and $$M_{107}$$.

(Pervushin's discovery of the primality of $$M_{61}$$ caused some to suggest that Mersenne's claim of the primality of $$M_{67}$$ may have been a copying error for $$M_{61}$$.)

Nobody will ever know how Mersenne came to his conclusions, as it is impossible with the mathematical knowledge of the time for him to have worked it all out by hand. The fact that he made so few mistakes is incredible.

The work continued, and does so to this day.


 * 1952: Raphael Robinson used a computer to show that $$M_{521}, M_{607}, M_{1279}, M_{2203}$$ and $$M_{2281}$$ are all prime.


 * During the next four decades, the count of known Mersenne primes was doubled by various mathematicians testing supercomputers.

Since then, hunting for Mersenne primes has become a casual hobby for anyone who has a computer.

Testing Primality of a Mersenne number
The Lucas-Lehmer Test is a way of determining the primality of a given $$M_p$$ without laboriously testing each possible prime divisor.

G.I.M.P.S.
"G.I.M.P.S." (Great Internet Mersenne Prime Search), or just "GIMPS", has become a gathering place for Number Theorists interested in the discovery of the Mersenne primes.

In August and September of 2008 alone, both the 45th and 46th Mersenne primes were discovered:

$$M_{43,112,609}$$ (a 12,978,189 digit number)

and

$$M_{37,156,667}$$ (an 11,185,272 digit number)

becoming the first Mersenne primes of 10 million digits to be found.

You can follow the work of G.I.M.P.S. at www.mersenne.org.

Currently known Mersenne Primes
Note that the index numbers of Mersenne primes after no. 39 are uncertain, as there may still be undiscovered Mersenne primes between the 39th and 40th. Not all numbers in that range have been explored yet.