
Glossary: Prime Pages: Top 5000: 
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
LucasLehmer 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)
Chris K. Caldwell © 19992018 (all rights reserved)
