Double Mersenne number

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

Double Mersenne primes
No. of known terms4
Conjectured no. of terms4
First terms7, 127, 2147483647
Largest known term170141183460469231731687303715884105727
OEIS index
  • A077586
  • a(n) = 2^(2^prime(n) − 1) − 1

where p is prime.

Examples

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

Double Mersenne primes

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 can be prime only if Mp is itself a Mersenne prime. For the first values of p for which Mp is prime, is known to be prime for p = 2, 3, 5, 7 while explicit factors of have been found for p = 13, 17, 19, and 31.

factorization of
23prime7
37prime127
531prime2147483647
7127prime170141183460469231731687303715884105727
11not primenot prime47 × 131009 × 178481 × 724639 × 2529391927 × 70676429054711 × 618970019642690137449562111 × ...
138191not prime338193759479 × 210206826754181103207028761697008013415622289 × ...
17131071not prime231733529 × 64296354767 × ...
19524287not prime62914441 × 5746991873407 × 2106734551102073202633922471 × 824271579602877114508714150039 × 65997004087015989956123720407169 × ...
23not primenot prime2351 × 4513 × 13264529 × 76899609737 × ...
29not primenot prime1399 × 2207 × 135607 × 622577 × 16673027617 × 4126110275598714647074087 × ...
312147483647not prime295257526626031 × 87054709261955177 × 242557615644693265201 × 178021379228511215367151 × ...
37not primenot prime
41not primenot prime
43not primenot prime
47not primenot prime
53not primenot prime
59not primenot prime
612305843009213693951unknown

Thus, the smallest candidate for the next double Mersenne prime is , 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 1×1036.[2] There are probably no other double Mersenne primes than the four known.[1][3]

Smallest prime factor of (where p is the nth prime) are

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1×1036) (sequence A309130 in the OEIS)

Catalan–Mersenne number conjecture

The recursively defined sequence

is called the sequence of Catalan–Mersenne numbers.[4] The first terms of the sequence (sequence A007013 in the OEIS) are:

Catalan discovered this sequence after the discovery of the primality of by Lucas in 1876.[1][5] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms 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 is not prime, there is a chance to discover this by computing modulo some small prime (using recursive modular exponentiation). If the resulting residue is zero, represents a factor of and thus would disprove its primality. Since is a Mersenne number, such a prime factor would have to be of the form . Additionally, because is composite when is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".

See also

References

  1. Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
  2. "Double Mersenne 61 factoring status". www.doublemersennes.org. Retrieved 31 March 2022.
  3. I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
  4. Weisstein, Eric W. "Catalan-Mersenne Number". MathWorld.
  5. "Questions proposées". Nouvelle correspondance mathématique. 2: 94–96. 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92:
    Prouver que 261  1 et 2127  1 sont des nombres premiers. (É. L.) (*).
    The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows:
    (*) Si l'on admet ces deux propositions, et si l'on observe que 22  1, 23  1, 27  1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2n  1 est un nombre premier p, 2p  1 est un nombre premier p', 2p'  1 est un nombre premier p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2n + 1 est un nombre premier. (E. C.)

Further reading

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