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 Mp 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 Mp is itself a Mersenne prime. For the first values of p for which Mp 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 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test. It has no prime factor below 4×1033.[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 M127) 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 MM127 is not prime, there is a chance to discover this by computing MM127 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”.