Editing Double Mersenne number (section)

== Double Mersenne primes ==
{{Infobox integer sequence
| name = Double Mersenne primes
| terms_number       = 4
| con_number         = 4
| first_terms        = 7, 127, 2147483647
| largest_known_term = 170141183460469231731687303715884105727
| OEIS               = A077586
| OEIS_name          = a(''n'') = 2^(2^prime(''n'') − 1) − 1
}}

A double Mersenne number that is [[prime number|prime]] is called a '''double Mersenne prime'''. Since a Mersenne number ''M''<sub>''p''</sub> can be prime only if ''p'' is prime, (see [[Mersenne prime]] for a proof), a double Mersenne number <math>M_{M_p}</math> can be prime only if ''M''<sub>''p''</sub> is itself a Mersenne prime. For the first values of ''p'' for which ''M''<sub>''p''</sub> is prime, <math>M_{M_{p}}</math> is known to be prime for ''p'' = 2, 3, 5, 7 while explicit factors of <math>M_{M_{p}}</math> have been found for ''p'' = 13, 17, 19, and mersenne prime 31.

{| class="wikitable"
|-
! <math>p</math> !! <math>M_{p} = 2^p-1</math> !! <math>M_{M_{p}} = 2^{2^p-1}-1</math> !! factorization of <math>M_{M_{p}}</math>
|-
| 2 || [[3 (number)|3]] || prime || 7
|-
| 3 || [[7 (number)|7]] || prime (triple) || 127
|-
| 5 || [[31 (number)|31]] || prime || 2147483647
|-
| 7 || [[127 (number)|127]] || prime (quadruple) || 170141183460469231731687303715884105727
|-
| 11 || not prime || not prime || 47 × 131009 × 178481 × 724639 × 2529391927 × 70676429054711 × 618970019642690137449562111 × ...
|-
| 13 || [[8191 (number)|8191]] || not prime || 338193759479 × 210206826754181103207028761697008013415622289 × ...
|-
| 17 || [[131071 (number)|131071]] || not prime || 231733529 × 64296354767 × ...
|-
| 19 || [[524287 (number)|524287]] || not prime || 62914441 × 5746991873407 × 2106734551102073202633922471 × 824271579602877114508714150039 × 65997004087015989956123720407169 × 4565880376922810768406683467841114102689 × ...
|-
| 23 || not prime || not prime || 2351 × 4513 × 13264529 × 76899609737 × ...
|-
| 29 || not prime || not prime || 1399 × 2207 × 135607 × 622577 × 16673027617 × 4126110275598714647074087 × ...
|-
| 31 || [[2147483647 (number)|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 (number)|2305843009213693951]] || unknown || 
|}

Thus, the smallest candidate for the next double Mersenne prime is <math>M_{M_{61}}</math>, or 2<sup>2305843009213693951</sup> − 1.
Being approximately 1.695{{e|694127911065419641}},
this number is far too large for any currently known [[primality test]]. It has no prime factor below 1&thinsp;×&thinsp;10<sup>36</sup>.<ref>{{cite web |title=Double Mersenne 61 factoring status |url=http://www.doublemersennes.org/mm61.php |website=www.doublemersennes.org |access-date=31 March 2022}}</ref>
There are probably no other double Mersenne primes than the four known.<ref name="Caldwell"/><ref>[https://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071444-6/S0025-5718-1955-0071444-6.pdf I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p.&nbsp;120-121] [retrieved 2012-10-19]</ref>

Smallest prime factor of <math>M_{M_{p}}</math> (where ''p'' is the ''n''th prime) are

:7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617, ... (next term is > 1&thinsp;×&thinsp;10<sup>36</sup>) {{OEIS|id=A309130}}