What is a Mersenne prime?

A Mersenne prime is a prime number that is one less than a power of two, expressed in the form 2 to the power of n minus one. Named after the seventeenth-century French friar Marin Mersenne, these numbers are closely connected to even perfect numbers through the Euclid-Euler theorem. Because numbers of this special form are easier to check for primality, many of the largest known prime numbers today are Mersenne primes.

What we know

• A Mersenne prime is a prime number of the form 2 to the power of n minus one for some integer n. [1, 2]
• Mersenne primes are named after Marin Mersenne, a French Minim friar who studied them in the early seventeenth century. [2]
• The Euclid-Euler theorem establishes a one-to-one correspondence between even perfect numbers and Mersenne primes. [2]
• Since 1997, all newly discovered Mersenne primes have been found by the Great Internet Mersenne Prime Search, a distributed computing project. [2]

Where to go next

• whoWho was Marin Mersenne?
• what caused itHow do perfect numbers relate to Mersenne primes?
• how it workedWhat is the Great Internet Mersenne Prime Search?
• the bigger pictureWhat are prime numbers and how do they work?
• compared withWhat is a Wieferich prime?
• an unexpected connectionHow do prime numbers connect to public-key cryptography?Prime numbers of special forms and large scales are fundamental to the routines in information technology like public-key cryptography, which relies on the difficulty of factoring large numbers.

Sources

• [1]Mersenne prime (Wikidata Q186875) · CC0 1.0
• [2]Mersenne prime · CC BY-SA 4.0
• [3]List of prime numbers · CC BY-SA 4.0
• [4]Prime number · CC BY-SA 4.0
• [5]Wieferich prime · CC BY-SA 4.0