Double Mersenne number
In mathematics, a double Mersenne number is a Mersenne number of the form
where p is prime.
Examples
The first four terms of the sequence of double Mersenne numbers are :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,, 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 | |||
| 2 | 3 | prime | 7 |
| 3 | 7 | prime | 127 |
| 5 | 31 | prime | 2147483647 |
| 7 | 127 | prime | 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 ×... |
| 23 | not prime | not prime | 2351 × 4513 × 13264529 × 76899609737 ×... |
| 29 | not prime | not prime | 1399 × 2207 × 135607 × 622577 × 16673027617 × 4126110275598714647074087 ×... |
| 31 | 2147483647 | not prime | 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 4×1033. There are probably no other double Mersenne primes than the four known.
Smallest prime factor of are
Catalan–Mersenne number conjecture
The recursively defined sequenceis called the Catalan–Mersenne numbers. The first terms of the sequence are:
Catalan came up with this sequence after the discovery of the primality of by Lucas in 1876. 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 simply because they are too huge. However, if is not prime, there is a chance to discover this by computing modulo some small prime . If the resulting residue is zero, represents a factor of and thus would disprove its primality. Since is a Mersenne number, such prime factor must 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.