Editing Double Mersenne number (section)

==Catalan–Mersenne number conjecture==

The [[recursion|recursively]] defined sequence
: <math>c_0 = 2</math>
: <math>c_{n+1} = 2^{c_n}-1 = M_{c_n}</math>
is called the sequence of '''Catalan–Mersenne numbers'''.<ref>{{MathWorld|urlname=Catalan-MersenneNumber|title=Catalan-Mersenne Number}}</ref> The first terms of the sequence {{OEIS|id=A007013}} are: 
:<math>c_0 = 2 </math>
:<math>c_1 = 2^2-1 = 3 </math>
:<math>c_2 = 2^3-1 = 7 </math>
:<math>c_3 = 2^7-1 = 127 </math>
:<math>c_4 = 2^{127}-1 = 170141183460469231731687303715884105727 </math>
:<math>c_5 = 2^{170141183460469231731687303715884105727}-1 \approx 5.45431 \times 10^{51217599719369681875006054625051616349} \approx 10^{10^{37.70942}}</math>
[[Eugène Charles Catalan|Catalan]] discovered this sequence after the discovery of the primality of <math>M_{127}=c_4</math> by [[Édouard Lucas|Lucas]] in 1876.<ref name="Caldwell"/><ref>{{cite journal|title=Questions proposées |journal=Nouvelle correspondance mathématique |volume=2 |year=1876 |pages=94–96 |url=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: {{quote|Prouver que 2<sup>61</sup>&nbsp;&minus;&nbsp;1 et 2<sup>127</sup>&nbsp;&minus;&nbsp;1 sont des nombres premiers. (É. L.) (*).}} The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows: {{quote|(*) Si l'on admet ces deux propositions, et si l'on observe que 2<sup>2</sup>&nbsp;&minus;&nbsp;1, 2<sup>3</sup>&nbsp;&minus;&nbsp;1, 2<sup>7</sup>&nbsp;&minus;&nbsp;1 sont aussi des nombres premiers, on a ce ''théorème empirique: Jusqu'à une certaine limite, si'' 2<sup>''n''</sup>&nbsp;&minus;&nbsp;1 ''est un nombre premier'' ''p'', 2<sup>''p''</sup>&nbsp;&minus;&nbsp;1 ''est un nombre premier'' ''p''<nowiki>'</nowiki>, 2<sup>''p''<nowiki>'</nowiki></sup>&nbsp;&minus;&nbsp;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, 2<sup>n</sup> + 1 est un nombre premier.'' (E. C.)}}</ref><ref>L. E. Dickson, ''[https://archive.org/details/historyoftheoryo01dick/ History of the theory of numbers. Volume 1: Divisibility and primality]'' (1919). Published by Washington, Carnegie Institution of Washington.</ref><sup>p. 22</sup> Catalan [[conjecture]]d 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 <math>c_5</math> is not prime, there is a chance to discover this by computing <math>c_5</math> [[Modular arithmetic|modulo]] some small prime <math>p</math> (using recursive [[modular exponentiation]]). If the resulting residue is zero, <math>p</math> represents a factor of <math>c_5</math> and thus would disprove its primality. Since <math>c_5</math> is a [[Mersenne number]], such a prime factor <math>p</math> would have to be of the form <math>2kc_4 +1</math>. Additionally, because <math>2^n-1</math> is [[composite number|composite]] when <math>n</math> is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.

If <math>c_5</math> were prime, it would also contradict the [[New Mersenne conjecture]]. It is known that <math>\frac{2^{c_4} + 1}{3}</math> is composite, with factor <math> 886407410000361345663448535540258622490179142922169401 = 5209834514912200c_4 + 1</math>.<ref name="Hoegge">[http://www.hoegge.dk/mersenne/NMC.html#unknown ''New Mersenne Conjecture'']</ref>