Double Mersenne number

Short description: Number of form 2^(2^p-1)-1 with prime exponent

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

MMp=22p−1−1

where p is prime.

## Contents

* 1 Examples
* 2 Double Mersenne primes
* 3 Catalan–Mersenne number conjecture
* 4 In popular culture
* 5 See also
* 6 References
* 7 Further reading
* 8 External links

## Examples

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

MM2=M3=7
MM3=M7=127
MM5=M31=2147483647
MM7=M127=170141183460469231731687303715884105727

## Double Mersenne primes

Double Mersenne primesNo. of known terms| 4

|

Conjectured no. of terms| 4
First terms| 7, 127, 2147483647
Largest known term| 170141183460469231731687303715884105727
OEIS index|

* A077586
* a(n) = 2^(2^prime(n) − 1) − 1

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

p | Mp=2p−1 | MMp=22p−1−1 | factorization of MMp

|
|
|

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 MM61, 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 MMp (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

c0=2
cn+1=2cn−1=Mcn

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

c0=2
c1=22−1=3
c2=23−1=7
c3=27−1=127
c4=2127−1=170141183460469231731687303715884105727
c5=2170141183460469231731687303715884105727−1≈5.45431×1051217599719369681875006054625051616349≈101037.70942

Catalan discovered this sequence after the discovery of the primality of M127=c4 by Lucas in 1876.[1][5][6]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 c5 is not prime, there is a chance to discover this by computing c5 modulo some small prime p (using recursive modular exponentiation). If the resulting residue is zero, p represents a factor of c5 and thus would disprove its primality. Since c5 is a Mersenne number, such a prime factor p would have to be of the form 2kc4+1. Additionally, because 2n−1 is composite when n is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

If c5 were prime, it would also contradict the New Mersenne conjecture. It is known that 2c4+13 is composite, with factor 886407410000361345663448535540258622490179142922169401=5209834514912200c4+1.[7]

## In popular culture

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number MM7 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

1. ↑ 1.0 1.1 1.2 Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
2. ↑ "Double Mersenne 61 factoring status". http://www.doublemersennes.org/mm61.php.
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". http://mathworld.wolfram.com/Catalan-MersenneNumber.html.
5. ↑ "Questions proposées". Nouvelle correspondance mathématique 2: 94–96. 1876. https://archive.org/stream/nouvellecorresp01mansgoog#page/n353/mode/2up. (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.)

6. ↑ L. E. Dickson, History of the theory of numbers. Volume 1: Divisibility and primality (1919). Published by Washington, Carnegie Institution of Washington.
7. ↑ New Mersenne Conjecture

## Further reading

* Dickson, L. E. (1971), History of the Theory of Numbers, New York: Chelsea Publishing .

## External links

* Weisstein, Eric W.. "Double Mersenne Number". http://mathworld.wolfram.com/DoubleMersenneNumber.html.
* Tony Forbes, A search for a factor of MM61 .
* Status of the factorization of double Mersenne numbers
* Double Mersennes Prime Search
* Operazione Doppi Mersennes

* v
* t
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Prime number classes

By formula|

* Fermat (22n \+ 1)
* Mersenne (2p − 1)
* Double Mersenne (22p−1 − 1)
* Wagstaff (2p \+ 1)/3
* Proth (k·2n \+ 1)
* Factorial (n! ± 1)
* Primorial (pn# ± 1)
* Euclid (pn# + 1)
* Pythagorean (4n \+ 1)
* Pierpont (2m·3n \+ 1)
* Quartan (x4 \+ y4)
* Solinas (2m ± 2n ± 1)
* Cullen (n·2n \+ 1)
* Woodall (n·2n − 1)
* Cuban (x3 − y3)/(x − y)
* Carol (2n − 1)2 − 2
* Kynea (2n \+ 1)2 − 2
* Leyland (xy \+ yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin
* Partitions
* Bell
* Motzkin

By property|

* Wieferich (pair)
* Wall–Sun–Sun
* Wolstenholme
* Wilson
* Lucky
* Fortunate
* Ramanujan
* Pillai
* Regular
* Strong
* Stern
* Supersingular (elliptic curve)
* Supersingular (moonshine theory)
* Good
* Super
* Higgs
* Highly cototient

Base-dependent|

* Happy
* Dihedral
* Palindromic
* Emirp
* Repunit (10n − 1)/9
* Permutable
* Circular
* Truncatable
* Strobogrammatic
* Minimal
* Weakly
* Full reptend
* Unique
* Primeval
* Self
* Smarandache–Wellin
* Tetradic

Patterns|

* Twin (p, p \+ 2)
* Bi-twin chain (n − 1, n \+ 1, 2n − 1, 2n \+ 1, …)
* Triplet (p, p \+ 2 or p \+ 4, p \+ 6)
* Quadruplet (p, p \+ 2, p \+ 6, p \+ 8)
* k−Tuple
* Cousin (p, p \+ 4)
* Sexy (p, p \+ 6)
* Chen
* Sophie Germain (p, 2p + 1)
* Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
* Safe (p, (p − 1)/2)
* Arithmetic progression (p \+ a·n, n = 0, 1, 2, 3, ...)
* Balanced (consecutive p − n, p, p \+ n)

By size|

* Titanic (1,000+ digits)
* Gigantic (10,000+ digits)
* Mega (1,000,000+ digits)
* Largest known

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

* Pseudoprime
* Catalan
* Elliptic
* Euler
* Euler–Jacobi
* Fermat
* Frobenius
* Lucas
* Somer–Lucas
* Strong
* Carmichael number
* Almost prime
* Semiprime
* Interprime
* Pernicious

Related topics|

* Probable prime
* Industrial-grade prime
* Illegal prime
* Formula for primes
* Prime gap

First 60 primes|

* 2
* 3
* 5
* 7
* 11
* 13
* 17
* 19
* 23
* 29
* 31
* 37
* 41
* 43
* 47
* 53
* 59
* 61
* 67
* 71
* 73
* 79
* 83
* 89
* 97
* 101
* 103
* 107
* 109
* 113
* 127
* 131
* 137
* 139
* 149
* 151
* 157
* 163
* 167
* 173
* 179
* 181
* 191
* 193
* 197
* 199
* 211
* 223
* 227
* 229
* 233
* 239
* 241
* 251
* 257
* 263
* 269
* 271
* 277
* 281

List of prime numbers

* v
* t
* e

Classes of natural numbers

Powers and
related numbers|

* Achilles
* Power of 2
* Power of 10
* Square
* Cube
* Fourth power
* Fifth power
* Sixth power
* Seventh power
* Perfect power
* Powerful
* Prime power

Of the form
a × 2b ± 1|

* Cullen
* Double Mersenne
* Fermat
* Mersenne
* Proth
* Thabit
* Woodall

Other polynomial
numbers|

* Carol
* Hilbert
* Idoneal
* Kynea
* Leyland
* Loeschian
* Lucky numbers of Euler

Recursively defined
numbers|

* Fibonacci
* Jacobsthal
* Leonardo
* Lucas
* Padovan
* Pell
* Perrin

Possessing a
specific set
of other numbers|

* Knödel
* Riesel
* Sierpinski

Expressible via
specific sums|

* Nonhypotenuse
* Polite
* Practical
* Primary pseudoperfect
* Ulam
* Wolstenholme

Figurate numbers|

| 2-dimensional|

| centered|

* Centered triangular
* Centered square
* Centered pentagonal
* Centered hexagonal
* Centered heptagonal
* Centered octagonal
* Centered nonagonal
* Centered decagonal
* Star

|

non-centered|

* Triangular
* Square
* Square triangular
* Pentagonal
* Hexagonal
* Heptagonal
* Octagonal
* Nonagonal
* Decagonal
* Dodecagonal

3-dimensional|

| centered|

* Centered tetrahedral
* Centered cube
* Centered octahedral
* Centered dodecahedral
* Centered icosahedral

|

non-centered|

* Tetrahedral
* Cube
* Octahedral
* Dodecahedral
* Icosahedral
* Stella octangula

pyramidal|

* Square pyramidal
* Pentagonal pyramidal
* Hexagonal pyramidal
* Heptagonal pyramidal

4-dimensional|

| centered|

* Centered pentachoric
* Squared triangular

|

non-centered|

* Pentatope
* 4-cube

Base-dependent
numbers|

| Arithmetic functions and dynamics|

* Digit sum
* Digital root
* Multiplicative
* Dudeney
* Factorion
* Happy
* Kaprekar
* Kaprekar's constant
* Keith
* Lychrel
* Meertens
* Narcissistic
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Persistence
* Additive
* Multiplicative
* Self
* Sum-product

|

P-adic numbers related|

* Automorphic

Divisor related|

* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith

Digit-composition related|

* Palindromic
* Pandigital
* Repdigit
* Repunit
* Self-descriptive
* Smarandache–Wellin
* Strictly non-palindromic
* Undulating

Digit-permutation related|

* Cyclic
* Digit-reassembly
* Parasitic
* Primeval
* Transposable

Other|

* Friedman
* Vampire

Combinatorial
numbers|

* Bell
* Cake
* Catalan
* Dedekind
* Delannoy
* Euler
* Fuss–Catalan
* Lazy caterer's sequence
* Lobb
* Motzkin
* Narayana
* Ordered Bell
* Schröder
* Schröder–Hipparchus

Pseudoprimes|

* Carmichael number
* Catalan pseudoprime
* Elliptic pseudoprime
* Euler pseudoprime
* Euler–Jacobi pseudoprime
* Fermat pseudoprime
* Frobenius pseudoprime
* Lucas pseudoprime
* Somer–Lucas pseudoprime
* Strong pseudoprime

Arithmetic functions|

| By properties of σ(n)|

* Abundant
* Arithmetic
* Colossally abundant
* Descartes
* Hemiperfect
* Highly abundant
* Highly composite
* Hyperperfect
* Multiply perfect
* Perfect
* Practical number
* Primitive abundant
* Quasiperfect
* Refactorable
* Sublime
* Superabundant
* Superior highly composite
* Superperfect
* Untouchable

|

By properties of Ω(n)|

* Almost prime
* Semiprime

By properties of φ(n)|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

By properties of s(n)|

* Almost perfect
* Amicable
* Betrothed
* Deficient
* Perfect
* Semiperfect
* Sociable

Other|

* Euclid
* Fortunate

Dividing a quotient|

* Wieferich
* Wall–Sun–Sun
* Wolstenholme prime
* Wilson

Other prime factor
or divisor related
numbers|

* Blum
* Erdős–Nicolas
* Erdős–Woods
* Friendly
* Giuga
* Harmonic divisor
* Lucas–Carmichael
* Pronic
* Regular
* Rough
* Smooth
* Sociable
* Sphenic
* Størmer
* Super-Poulet
* Zeisel

Binary numbers|

* Evil
* Odious
* Pernicious

Generated
via a sieve|

* Lucky
* Prime

Sorting related|

* Pancake number
* Sorting number

Natural language related|

* Aronson's sequence
* Ban

Graphemics related|

* Strobogrammatic

* Mathematics portal

* v
* t
* e

Marin Mersenne

* Mersenne conjectures
* Mersenne's laws
* Mersenne prime
* Double Mersenne number
* Great Internet Mersenne Prime Search
* Mersenne Twister

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