Number of known terms | 4 |
---|---|

Conjectured number of terms | 4 |

First terms | 7, 127, 2147483647 |

Largest known term | 170141183460469231731687303715884105727 |

OEIS index | A077586 |

In mathematics, a **double Mersenne number** is a Mersenne number of the form

- {\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}

where *p* is a prime exponent.

## Examples[edit]

The first four terms of the sequence of double Mersenne numbers are^{[1]} (sequence A077586 in the OEIS):

- {\displaystyle M_{M_{2}}=M_{3}=7}
- {\displaystyle M_{M_{3}}=M_{7}=127}
- {\displaystyle M_{M_{5}}=M_{31}=2147483647}
- {\displaystyle M_{M_{7}}=M_{127}=170141183460469231731687303715884105727}

## Double Mersenne primes[edit]

A double Mersenne number that is prime is called a **double Mersenne prime**. Since a Mersenne number *M*_{p} can be prime only if *p* is prime, (see Mersenne prime for a proof), a double Mersenne number {\displaystyle M_{M_{p}}} can be prime only if *M*_{p} is itself a Mersenne prime. For the first values of *p* for which *M*_{p} is prime, {\displaystyle M_{M_{p}}} is known to be prime for *p* = 2, 3, 5, 7 while explicit factors of {\displaystyle M_{M_{p}}} have been found for *p* = 13, 17, 19, and 31.

{\displaystyle p} | {\displaystyle M_{p}=2^{p}-1} | {\displaystyle M_{M_{p}}=2^{2^{p}-1}-1} |
---|---|---|

2 | 3 | prime |

3 | 7 | prime |

5 | 31 | prime |

7 | 127 | prime |

11 | not prime | — |

13 | 8191 | not prime |

17 | 131071 | not prime |

19 | 524287 | not prime |

23 | not prime | — |

29 | not prime | — |

31 | 2147483647 | not prime |

37 | not prime | — |

41 | not prime | — |

43 | not prime | — |

47 | not prime | — |

53 | not prime | — |

59 | not prime | — |

61 | 2305843009213693951 | unknown |

Thus, the smallest candidate for the next double Mersenne prime is {\displaystyle M_{M_{61}}}, or 2^{2305843009213693951} − 1. Being approximately 1.695×10^{694127911065419641}, this number is far too large for any currently known primality test. It has no prime factor below 4×10^{33}.^{[2]} There are probably no other double Mersenne primes than the four known.^{[1]}^{[3]}

## Catalan–Mersenne number conjecture[edit]

Write {\displaystyle M(p)} instead of {\displaystyle M_{p}}. A special case of the double Mersenne numbers, namely the recursively defined sequence

is called the **Catalan–Mersenne numbers**.^{[4]} Catalan came up with this sequence after the discovery of the primality of *M*(127) = *M*(*M*(*M*(*M*(2)))) by Lucas in 1876.^{[1]}^{[5]} Catalan conjectured that they are prime “up to a certain limit”. Although the first five terms (below *M*_{127}) are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if *M*_{M127} is not prime, there is a chance to discover this by computing *M*_{M127} modulo some small prime *p* (using recursive modular exponentiation).^{[6]}

## In popular culture[edit]

In the Futurama movie *The Beast with a Billion Backs*, the double Mersenne number {\displaystyle M_{M_{7}}} is briefly seen in “an elementary proof of the Goldbach conjecture“. In the movie, this number is known as a “martian prime”.