Mersenne

Mersenne is a surname most often associated with Marin Mersenne (c. 1588–1648), a French Minim friar and mathematician. The name also designates two closely related concepts in number theory: Mersenne numbers and Mersenne primes, named after him.

Marin Mersenne was active in the early 17th century and contributed to the study of prime numbers. He investigated primes of the form 2^p − 1 and compiled lists of candidates, helping to popularize the idea that certain exponential forms yield primes. His work laid groundwork for later systematic study of these primes.

A Mersenne number is defined as M_p = 2^p − 1 for a positive integer p. If p is prime and M_p is prime, then M_p is called a Mersenne prime. These primes are intimately connected to even perfect numbers: Euler showed that every even perfect number has the form 2^{p−1}(2^p − 1) with M_p prime, and the existence of such primes yields corresponding even perfect numbers.

Primality testing for M_p uses specialized methods, notably the Lucas–Lehmer test, which is efficient for this particular form. Because Mersenne primes can be extremely large, their discovery relies heavily on computer-aided search, most famously through the Great Internet Mersenne Prime Search (GIMPS).

As of 2018, the largest known Mersenne prime has more than 24 million digits. The study of Mersenne primes remains active, with new primes discovered periodically and implications for number theory and computational mathematics.