# Mersenne prime

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### Etymology

Named after Marin Mersenne (1588–1648), French theologian, philosopher, mathematician, and music theorist.

### Pronunciation

• IPA(key): /mɛə(ɹ)ˈsɛn ˈpɹaɪm/

### Noun

Mersenne prime (plural Mersenne primes)

1. (number theory) A prime number which is one less than a power of two (i.e., is expressible in the form 2n – 1; for example, 31 = 25 – 1).
• 2004, Sheldon Axler, 3: Mathematicians Versus the Silicon Age: Who Wins?, David F. Hayes, Tatiana Shubin (editors), Mathematical Adventures for Students and Amateurs, Mathematical Society of America, page 20,
In addition to being the largest known Mersenne prime, this number is also currently the largest known prime number of any type.
• 2005, Jean-Claude Bajard, Laurent Imbert, Thomas Plantard, Modular Number Systems: Beyond the Mersenne Family, Helena Handschuh, M. Anwar Hasan (editors), Selected Areas in Cryptography: 11th International Workshop, SAC 2004, Revised Selected Papers, Springer, LNCS 3357, page 159,
Mersenne numbers of the form $2^{m}-1$ are well known examples, but they are not useful for cryptography because there are only a few primes (the first Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, etc).
• 2013, Louis Komzsik, Magnificent Seven: The Happy Number, Trafford Publishing, page 71,
Mersenne primes, named after the French priest first describing them in the 17th century, are of the form of $2^{p}-1$ , where the exponent $p$ is a prime itself. Obviously $7$ is a Mersenne prime, since $2^{3}-1=7$ , and $3$ is a prime. The interesting twist comes in the fact that there are certain primes that are double Mersenne primes. These are of the form $2^{2^{p}-1}-1$ , meaning that the exponent now is not just a prime, but it is itself a Mersenne prime. Since $2^{2^{2}-1}-1=7$ , $7$ is the very first double Mersenne prime.