Mersenne prime

A Mersenne number 2ⁿ-1 which is prime is called a Mersenne prime. If m divides n, then 2^m-1 divides 2ⁿ-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 M₆₇ and M₂₅₇ being prime, and he omitted M₆₁, M₈₉, and M₁₀₇. 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).>/p>

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