Double Mersenne number

In number theory , a double Mersenne number is a number of the form , where a natural number and the -th is Mersenne number . ${\displaystyle M_{M_{n}}=2^{2^{n}-1}-1}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle M_{n}}$ ${\displaystyle n}$

Examples

The first five double Mersenne numbers are the following (sequence A077585 in OEIS ):

${\displaystyle {\begin{aligned}M_{M_{1}}&=&2^{2^{1}-1}-1&=&2^{1}-1&=&M_{1}&=&1\\M_{M_{2}}&=&2^{2^{2}-1}-1&=&2^{3}-1&=&M_{3}&=&7\\M_{M_{3}}&=&2^{2^{3}-1}-1&=&2^{7}-1&=&M_{7}&=&127\\M_{M_{4}}&=&2^{2^{4}-1}-1&=&2^{15}-1&=&M_{15}&=&32767\\M_{M_{5}}&=&2^{2^{5}-1}-1&=&2^{31}-1&=&M_{31}&=&2147483647\\\end{aligned}}}$

properties

Every double Mersenne number is also a Mersenne number. ${\displaystyle M_{M_{n}}}$

Proof:

Be . Then: ${\displaystyle k:=2^{n}-1}$

${\displaystyle M_{M_{n}}=2^{2^{n}-1}-1=2^{k}-1=M_{k}}$

Thus the double Mersenne number is also the th Mersenne number , which was to be shown. ${\displaystyle M_{M_{n}}}$ ${\displaystyle k}$ ${\displaystyle M_{k}}$ ${\displaystyle \Box }$

Double Mersenne primes

If a double Mersenne number is a prime number, it is called a double Mersenne prime number . ${\displaystyle M_{M_{p}}}$

Examples

The first four double Mersenne prime numbers are the following (sequence A077586 in OEIS ):

${\displaystyle {\begin{aligned}M_{M_{2}}&=&2^{2^{2}-1}-1&=&2^{3}-1&=&M_{3}&=&7\\M_{M_{3}}&=&2^{2^{3}-1}-1&=&2^{7}-1&=&M_{7}&=&127\\M_{M_{5}}&=&2^{2^{5}-1}-1&=&2^{31}-1&=&M_{31}&=&2147483647\\M_{M_{7}}&=&2^{2^{7}-1}-1&=&2^{127}-1&=&M_{127}&=&170141183460469231731687303715884105727\end{aligned}}}$

More than these four are currently unknown.

properties

Be with natural . Then: ${\displaystyle M_{M_{n}}=2^{2^{n}-1}-1}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle M_{M_{n}}=2^{2^{n}-1}-1}$is only a prime number if the Mersenne number is also a prime number. ${\displaystyle M_{n}}$

The converse is not true: if is a prime, it may or may not be prime. ${\displaystyle M_{n}}$ ${\displaystyle M_{M_{n}}=2^{2^{n}-1}-1}$

table

The following table indicates which double Mersenne numbers with are prime which is not and which is not even known whether it is prime or not. Here is a -digit composite number and a -digit remainder factor: ${\displaystyle M_{M_{p}}}$ ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle Z_{k}}$ ${\displaystyle k}$ ${\displaystyle R_{k}}$ ${\displaystyle k}$

${\displaystyle p}$	${\displaystyle M_{p}=2^{p}-1}$	${\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}$	Number of digits from ${\displaystyle M_{M_{p}}}$	Prime number?	Factorization of ${\displaystyle M_{M_{p}}}$
2	${\displaystyle M_{2}=2^{2}-1=3\in \mathbb {P} }$	${\displaystyle M_{M_{2}}=2^{3}-1=7}$	1	prim	${\displaystyle 7}$
3	${\displaystyle M_{3}=2^{3}-1=7\in \mathbb {P} }$	${\displaystyle M_{M_{3}}=2^{7}-1=127}$	3	prim	${\displaystyle 127}$
5	${\displaystyle M_{5}=2^{5}-1=31\in \mathbb {P} }$	${\displaystyle M_{M_{5}}=2^{31}-1=2147483647}$	10	prim	${\displaystyle 2147483647}$
7th	${\displaystyle M_{7}=2^{7}-1=127\in \mathbb {P} }$	${\displaystyle M_{M_{7}}=2^{127}-1}$	39	prim	${\displaystyle 170141183460469231731687303715884105727}$
11	${\displaystyle M_{11}=2^{11}-1=2047\not \in \mathbb {P} }$	${\displaystyle M_{M_{11}}=2^{2047}-1}$	617	not prime	${\displaystyle 47\cdot 131009\cdot 178481\cdot 724639\cdot 2529391927\cdot 70676429054711\cdot 618970019642690137449562111\cdot Z_{549}}$
13	${\displaystyle M_{13}=2^{13}-1=8191\in \mathbb {P} }$	${\displaystyle M_{M_{13}}=2^{8191}-1}$	2,466	not prime	${\displaystyle 338193759479\cdot 210206826754181103207028761697008013415622289\cdot Z_{2410}}$
17th	${\displaystyle M_{17}=2^{17}-1=131071\in \mathbb {P} }$	${\displaystyle M_{M_{17}}=2^{131071}-1}$	39,457	not prime	${\displaystyle 231733529\cdot 64296354767\cdot Z_{39438}}$
19th	${\displaystyle M_{19}=2^{19}-1=524287\in \mathbb {P} }$	${\displaystyle M_{M_{19}}=2^{524287}-1}$	157,827	not prime	${\displaystyle 62914441\cdot 5746991873407\cdot 2106734551102073202633922471\cdot 824271579602877114508714150039\cdot 65997004087015989956123720407169\cdot Z_{157717}}$
23	${\displaystyle M_{23}=2^{23}-1=8388607\not \in \mathbb {P} }$	${\displaystyle M_{M_{23}}=2^{8388607}-1}$	2,525,223	not prime	${\displaystyle 2351\cdot 4513\cdot 13264529\cdot 76899609737\cdot Z_{2525198}}$
29	${\displaystyle M_{29}=2^{29}-1=536870911\not \in \mathbb {P} }$	${\displaystyle M_{M_{29}}=2^{536870911}-1}$	161.614.249	not prime	${\displaystyle 1399\cdot 2207\cdot 135607\cdot 622577\cdot 16673027617\cdot 4126110275598714647074087\cdot R_{161614196}}$
31	${\displaystyle M_{31}=2^{31}-1=2147483647\in \mathbb {P} }$	${\displaystyle M_{M_{31}}=2^{2147483647}-1}$	646.456.993	not prime	${\displaystyle 295257526626031\cdot 87054709261955177\cdot 242557615644693265201\cdot 178021379228511215367151\cdot R_{646456918}}$
37	${\displaystyle M_{37}=2^{37}-1=137438953471\not \in \mathbb {P} }$	${\displaystyle M_{M_{37}}=2^{137438953471}-1}$	41,373,247,568	not prime	unknown
41	${\displaystyle M_{41}=2^{41}-1=2199023255551\not \in \mathbb {P} }$	${\displaystyle M_{M_{41}}=2^{2199023255551}-1}$	661.971.961.084	not prime	unknown
43	${\displaystyle M_{43}=2^{43}-1=8796093022207\not \in \mathbb {P} }$	${\displaystyle M_{M_{43}}=2^{8796093022207}-1}$	2,647,887,844,335	not prime	unknown
47	${\displaystyle M_{47}=2^{47}-1=140737488355327\not \in \mathbb {P} }$	${\displaystyle M_{M_{47}}=2^{140737488355327}-1}$	42.366.205.509.364	not prime	unknown
53	${\displaystyle M_{53}=2^{53}-1=9007199254740991\not \in \mathbb {P} }$	${\displaystyle M_{M_{53}}=2^{9007199254740991}-1}$	2,711,437,152,599,296	not prime	unknown
59	${\displaystyle M_{59}=2^{59}-1=576460752303423487\not \in \mathbb {P} }$	${\displaystyle M_{M_{59}}=2^{576460752303423487}-1}$	173,531,977,766,354,911	not prime	unknown
61	${\displaystyle M_{61}=2^{61}-1=2305843009213693951\in \mathbb {P} }$	${\displaystyle M_{M_{61}}=2^{2305843009213693951}-1}$	694.127.911.065.419.642	unknown	no prime factor ${\displaystyle p<4\cdot 10^{33}}$

The double Mersenne number is far too large to be applied to a well-known prime number test (especially the Lucas-Lehmer test tailored to Mersenne numbers ). So you don't even know if it's compound or not. For all other prime numbers one also does not yet know whether it is prime or not. It is believed , however , that there are no other double Mersenne primes other than the first four. ${\displaystyle M_{M_{61}}}$ ${\displaystyle p>61}$ ${\displaystyle M_{M_{p}}}$

Catalan-Mersenne numbers

The following recursively defined numbers are called Catalan-Mersenne numbers (sequence A007013 in OEIS ):

${\displaystyle {\begin{aligned}C_{0}&=&2\\C_{1}&=&M(2)&=&M_{2}&=&2^{C_{0}}-1&=&2^{2}-1&=M_{2}=3\\C_{2}&=&M(M(2))&=&M_{M_{2}}&=&2^{C_{1}}-1&=&2^{2^{2}-1}-1&=M_{3}=7\\C_{3}&=&M(M(M(2)))&=&M_{M_{M_{2}}}&=&2^{C_{2}}-1&=&2^{2^{2^{2}-1}-1}-1&=M_{7}=127\\C_{4}&=&M(M(M(M(2))))&=&M_{M_{M_{M_{2}}}}&=&2^{C_{3}}-1&=&2^{2^{2^{2^{2}-1}-1}-1}-1&=M_{127}=170141183460469231731687303715884105727\\C_{5}&=&M(M(M(M(M(2)))))&=&M_{M_{M_{M_{M_{2}}}}}&=&2^{C_{4}}-1&=&2^{2^{2^{2^{2^{2}-1}-1}-1}-1}-1&=M_{170141183460469231731687303715884105727}=\ldots \\\ldots \\C_{n}&=&\ldots &=&\ldots &=&2^{C_{n-1}}-1\end{aligned}}}$

One does not know whether it is prime or not because it is much too large (much larger than , which is already much too large for known primality tests; it has 51,217,599,719,369,681,875,006,054,625,051,616,350 digits ). All that is known is that it has no prime factor . However, it is believed that this number is composed. But when composed, would any further with also composed, as already previously been shown that (and is a double Mersenne number) is prime only if also a prime number. ${\displaystyle C_{5}}$ ${\displaystyle M_{M_{61}}}$ ${\displaystyle p<5\cdot 10^{51}}$ ${\displaystyle C_{5}}$ ${\displaystyle C_{5}}$ ${\displaystyle C_{n}}$ ${\displaystyle n\geq 6}$ ${\displaystyle M_{C_{n}}}$ ${\displaystyle C_{n}}$ ${\displaystyle C_{n}}$

The mathematician Eugène Charles Catalan first studied these numbers after proving the primality of Édouard Lucas in 1876. He was the first to claim that these numbers are all prime up to a certain upper limit and then put all others together. ${\displaystyle M(M(M(M(2))))=M_{127}}$ ${\displaystyle C_{n}}$

properties

The set of Catalan-Mersenne numbers are a subset of the set of double Mersenne numbers. In other words: every Catalan-Mersenne number is also a double Mersenne number.

Trivia

In the series Futurama , the double Mersenne number appears in the episode The Era of the Tentacle (2008). It appears briefly in the background on a board in an "elementary proof of Goldbach's conjecture " (which in reality has not yet been proven). In this episode, that number is referred to as the martian prime . ${\displaystyle M_{M_{7}}}$

Web links

• Eric W. Weisstein : Double Mersenne Number . In: MathWorld (English).
• Eric W. Weisstein : Catalan-Mersenne Number . In: MathWorld (English).
• Double Mersennes Prime Search

Individual evidence

1. ↑ MM ₆₁ - A search for a factor of 2 ^{2 61 -1} -1
2. ↑ MM ₆₁ - A search for a factor of 2 ^{2 61 -1} -1 - Lists
3. ↑ ^{a b} Chris K. Caldwell: Mersenne Primes: History, Theorems and Lists - Conjectures and Unsolved Problems. Prime Pages, accessed December 25, 2018 . 
4. ^ IJ Good: Conjectures concerning the Mersenne numbers. (PDF) Mathematics of Computation 9 , 1955, pp. 120–121 , accessed December 25, 2018 . 
5. ↑ ^{a b c} Eric W. Weisstein : Catalan-Mersenne Number . In: MathWorld (English).
6. ↑ Landon Curt Noll: Landon Curt Noll's prime pages. Retrieved December 26, 2018 . 
7. ^ Eugène Charles Catalan : Nouvelle correspondance mathématique - Questions proposées. Imprimeur de l'academie royale de Belgique, 1878, pp. 94-96 , accessed on December 26, 2018 (French, question 92). 
8. ↑ Les mathématiques de Futurama - Grands théorèmes. Retrieved December 26, 2018 (French).