Mersenne prime

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

• IPA^(key): /mɛə(ɹ)ˈsɛn ˈpɹaɪm/

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 ${\displaystyle 2^{n}-1}$; for example, ${\displaystyle 31=2^{5}-1}$).

   Coordinate terms: Fermat prime, Sophie Germain prime

   • 2004, Sheldon Axler, “3: Mathematicians Versus the Silicon Age: Who Wins?”, in David F. Hayes, Tatiana Shubin, editors, Mathematical Adventures for Students and Amateurs, American Mathematical Society, 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 ${\displaystyle 2^{m}-1}$ 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 ${\displaystyle 2^{p}-1}$, where the exponent ${\displaystyle p}$ is a prime itself. Obviously ${\displaystyle 7}$ is a Mersenne prime, since ${\displaystyle 2^{3}-1=7}$, and ${\displaystyle 3}$ 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 ${\displaystyle 2^{2^{p}-1}-1}$, meaning that the exponent now is not just a prime, but it is itself a Mersenne prime. Since ${\displaystyle 2^{2^{2}-1}-1=7}$, ${\displaystyle 7}$ is the very first double Mersenne prime.

• Mersenne number
• Mersenne twister

• perfect number

• Fermat number on Wikipedia.Wikipedia
• Fermat's little theorem on Wikipedia.Wikipedia
• Great Internet Mersenne Prime Search on Wikipedia.Wikipedia
• Perfect number on Wikipedia.Wikipedia
• Euclid–Euler theorem on Wikipedia.Wikipedia