Double Mersenne number

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

M_{M_{p}}=2^{2^{p}-1}-1

where p is prime.

YouTube Encyclopedic

1/5
Views:
302 766
358 449
7 834
5 187
559 709

Transcription

Examples

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

M_{M_{2}}=M_{3}=7

M_{M_{3}}=M_{7}=127

M_{M_{5}}=M_{31}=2147483647

M_{M_{7}}=M_{127}=170141183460469231731687303715884105727

Double Mersenne primes

Double Mersenne primes
No. 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 M_p can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number $M_{M_{p}}$ can be prime only if M_p is itself a Mersenne prime. For the first values of p for which M_p is prime, $M_{M_{p}}$ is known to be prime for p = 2, 3, 5, 7 while explicit factors of $M_{M_{p}}$ have been found for p = 13, 17, 19, and 31.

$p$	$M_{p}=2^{p}-1$	$M_{M_{p}}=2^{2^{p}-1}-1$	factorization of $M_{M_{p}}$
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 $M_{M_{61}}$ , or 2^{2305843009213693951} − 1. Being approximately 1.695×10^{694127911065419641}, 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 $M_{M_{p}}$ (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

c_{0}=2

c_{n+1}=2^{c_{n}}-1=M_{c_{n}}

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

c_{0}=2

c_{1}=2^{2}-1=3

c_{2}=2^{3}-1=7

c_{3}=2^{7}-1=127

c_{4}=2^{127}-1=170141183460469231731687303715884105727

c_{5}=2^{170141183460469231731687303715884105727}-1\approx 5.45431\times 10^{51217599719369681875006054625051616349}\approx 10^{10^{37.70942}}

Catalan discovered this sequence after the discovery of the primality of $M_{127}=c_{4}$ by Lucas in 1876.^[1]^[5] 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 $c_{5}$ is not prime, there is a chance to discover this by computing $c_{5}$ modulo some small prime $p$ (using recursive modular exponentiation). If the resulting residue is zero, $p$ represents a factor of $c_{5}$ and thus would disprove its primality. Since $c_{5}$ is a Mersenne number, such a prime factor $p$ would have to be of the form $2kc_{4}+1$ . Additionally, because $2^{n}-1$ 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.

In popular culture

In the Futurama movie The Beast with a Billion Backs, the double Mersenne number $M_{M_{7}}$ is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".

References

^ ^a ^b ^c Chris Caldwell, Mersenne Primes: History, Theorems and Lists at the Prime Pages.
^ "Double Mersenne 61 factoring status". www.doublemersennes.org. Retrieved 31 March 2022.
^ I. J. Good. Conjectures concerning the Mersenne numbers. Mathematics of Computation vol. 9 (1955) p. 120-121 [retrieved 2012-10-19]
^ Weisstein, Eric W. "Catalan-Mersenne Number". MathWorld.
^ "Questions proposées". Nouvelle correspondance mathématique. 2: 94–96. 1876. (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, 2^p − 1 est un nombre premier p', 2^p' − 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.)

External links

Weisstein, Eric W. "Double Mersenne Number". MathWorld.
Tony Forbes, A search for a factor of MM61 Archived 2009-02-08 at the Wayback Machine.
Status of the factorization of double Mersenne numbers
Double Mersennes Prime Search
Operazione Doppi Mersennes

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
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 Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number 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

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

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 Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

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

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
- Multiplicative

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

P-adic numbers-related

Automorphic
- Trimorphic

Digit-composition related

Digit-permutation related

Divisor-related

Other

Friedman

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

This page was last edited on 11 March 2024, at 04:30

From Wikipedia, the free encyclopedia