# Mersenne prime

A Mersenne number 2^{n}-1 which is prime is called a
**Mersenne prime**. If *m* divides *n*, then
2^{m}-1 divides 2^{n}-1, so a Mersenne
prime has a prime exponent. However, very few of the numbers of the
form 2^{p}-1 (*p* prime) are prime. Mersenne
Numbers are the easiest type of number to prove prime (because of the
Lucas-Lehmer test), so are usually the largest primes on the list of
largest known primes).

Primes of this form were
first studied by Euclid who explored their relationship
with the even perfect numbers. They were named after
Mersenne because he wrote to so many mathematicians
encouraging their study and because he sparked the
interest of generations of mathematicians by claiming
in 1644 that
M_{p} was prime for 2, 3, 5, 7, 13, 17, 19,
31, 67, 127, 257 and for no other primes *p*
less than 257. It took three centuries to completely
test his bold claim, and when done, it was discovered
that he was wrong about M_{67} and
M_{257} being prime, and he omitted
M_{61}, M_{89}, and M_{107}.
See the entry on Mersenne's conjecture for more information.

Mersenne primes (sometimes just called
**Mersennes**) are also generalized repunit primes
and trivial circular primes (radix two).

**See Also:** Cullens, FermatNumber

**Related pages** (outside of this work)

- Mersenne Primes (definitions, theorems, records, and history)
- The great Internet Mersenne prime search
- Proof of the Lucas-Lehmer Test
- The twenty largest Mersenne primes