Double Mersenne number

From Wikipedia, the free encyclopedia
Double Mersenne primes
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}{\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}M_{M_2} = M_3 = 7
{\displaystyle M_{M_{3}}=M_{7}=127}M_{M_3} = M_7 = 127
{\displaystyle M_{M_{5}}=M_{31}=2147483647}M_{M_5} = M_{31} = 2147483647
{\displaystyle M_{M_{7}}=M_{127}=170141183460469231731687303715884105727}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}}}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}}}{\displaystyle M_{M_{p}}} is known to be prime for p = 2, 3, 5, 7 while explicit factors of {\displaystyle M_{M_{p}}}{\displaystyle M_{M_{p}}} have been found for p = 13, 17, 19, and 31.

{\displaystyle p}p {\displaystyle M_{p}=2^{p}-1}{\displaystyle M_{p}=2^{p}-1} {\displaystyle M_{M_{p}}=2^{2^{p}-1}-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}}}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)}M(p) instead of {\displaystyle M_{p}}M_{p}. A special case of the double Mersenne numbers, namely the recursively defined sequence

2, M(2), M(M(2)), M(M(M(2))), M(M(M(M(2)))), … (sequence A007013 in the OEIS)

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}}}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”.

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