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A Mersenne number 2n-1 which is prime is called a Mersenne prime. If m divides n, then 2m-1 divides 2n-1, so a Mersenne prime has a prime exponent. However, very few of the numbers of the form 2p-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 Mp 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 M67 and M257 being prime, and he omitted M61, M89, and M107. See the entry on Mersenne's conjecture for more information.
Related pages (outside of this work)