In mathematics, a double Mersenne number is a Mersenne number of the form where is prime.
Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages. :
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, and 7 while explicit factors of have been found for p = 13, 17, 19, and 31.
factorization of 2 3 prime 7 3 7 prime (triple) 127 5 31 prime 2147483647 7 127 prime (quadruple) 170141183460469231731687303715884105727 11 not prime not prime 47 × 131009 × 178481 × 724639 × 2529391927 × 70676429054711 × 618970019642690137449562111 × ... 13 8191 not prime 338193759479 × 210206826754181103207028761697008013415622289 × ... 17 131071 not prime 231733529 × 64296354767 × ... 19 524287 not prime 62914441 × 5746991873407 × 2106734551102073202633922471 × 824271579602877114508714150039 × 65997004087015989956123720407169 × 4565880376922810768406683467841114102689 × ... 23 not prime not prime 2351 × 4513 × 13264529 × 285212639 × 76899609737 × ... 29 not prime not prime 1399 × 2207 × 135607 × 622577 × 16673027617 × 52006801325877583 × 4126110275598714647074087 × ... 31 2147483647 not prime (triple mersenne number) 295257526626031 × 87054709261955177 × 242557615644693265201 × 178021379228511215367151 × ... 37 not prime not prime 41 not prime not prime 43 not prime not prime 47 not prime not prime 53 not prime not prime 59 not prime not prime 61 2305843009213693951 unknown
Thus, the smallest candidate for the next double Mersenne prime is , or 22305843009213693951 − 1. Being approximately 1.695, this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036. There are probably no other double Mersenne primes than the four known.I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
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)
Catalan–Mersenne number conjecture
The recursively defined sequence
is called the sequence of Catalan–Mersenne numbers. The first terms of the sequence are:
Catalan discovered this sequence after the discovery of the primality of by Lucas in 1876. (probably collected by the editor). Almost all of the questions are signed by Édouard Lucas as is number 92: The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows: L. E. Dickson, History of the theory of numbers. Volume 1: Divisibility and primality (1919). Published by Washington, Carnegie Institution of Washington.p. 22 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.
If were prime, it would also contradict the New Mersenne conjecture. It is known that is composite, with factor .New Mersenne Conjecture
In popular culture 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 Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime
References
Further reading .
External links
Tony Forbes, A search for a factor of MM61 . Status of the factorization of double Mersenne numbers Double Mersennes Prime Search Operazione Doppi Mersennes