Mersenne number

A Mersenne number (named after French monk ) is a number of the form \(M_n = 2^n - 1\).Wolfram MathWorld. Mersenne Number
When \(M_n\) is prime, it is called Mersenne prime. In order for \(M_n\) to be prime, n must itself be prime.Wolfram MathWorld. Mersenne Prime As of September 2024, there are 51 known Mersenne primes, with \(M_{82589933} = 2^{82589933} - 1\) being the largest.Great Internet Mersenne Prime Search GIMPS. List of Known Mersenne Prime Numbers Retrieved 2024-09-17.
A double Mersenne number is a Mersenne number whose exponent is a Mersenne prime.Wolfram MathWorld. Double Mersenne Number Retrieved 2024-09-17 As of 2024, it is known that the first four double Mersenne numbers, \(M_{M_2}\), \(M_{M_3}\), \(M_{M_5}\), \(M_{M_7}\), are prime and the next four, \(M_{M_{13}}\), \(M_{M_{17}}\), \(M_{M_{19}}\), \(M_{M_{31}}\) are composite. The primality status of other known double Mersenne numbers remain unknown.
A Catalan-Mersenne number (named after ) is a number in the following sequence:Wolfram MathWorld. Catalan-Mersenne NumberOEIS A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.
2, \(M_2\), \(M_{M_2}\), \(M_{M_{M_2}}\), \(M_{M_{M_{M_2}}}\), \(M_{M_{M_{M_{M_2}}}}\), ...
The first 5 terms are prime. The sixth term, \(M_{M_{M_{M_{M_2}}}}=M_{M_{127}}\), is far too large for any known primality test. It is equal to \(2^{170,141,183,460,469,231,731,687,303,715,884,105,727}-1\), which has exactly 51,217,599,719,369,681,875,006,054,625,051,616,351 (more than fifty-one undecillion) decimal digits. It is only possible to prove that it is composite if it has a factor small enough to be discovered.
== Examples ==
*The first 3 Mersenne primes are 3, 7, 31.
*The 4th Mersenne prime 127 (one hundred twenty-seven) is a positive integer equal to \(2^{2^3-1}-1\). It is also a Catalan-Mersenne number. In many progamming languages it is the maximum value of an 8-bit signed integer.Microsoft. Integral numeric types (C# reference) Retrieved 2024-09-17.
*2,047 is the smallest composite Mersenne number with prime index, in this case, (211−1). The next Mersenne number with prime index, however, which is 213−1 or 8,191, is prime.
**It is also the smallest to base 2.OEIS. A001262 Strong pseudoprimes to base 2. Retrieved 2024-09-17.
*The 5th Mersenne prime \(8,191=2^{13}-1\) is the largest known number which is a repunit with at least three digits in more than one base. The states that 31 and 8,191 are the only two numbers with this property.Goormaghtigh conjecture
*The 6th Mersenne prime is 217−1 = 131,071.
*The 7th Mersenne prime is 219−1 = 524,287.
*The 8th Mersenne prime, \(2,147,483,647 = 2^{31} - 1 = 2^{2^5 - 1} - 1\), is the maximum value of a 32-bit signed integer, which have the range [-2147483648, 2147483647]. It is also the largest known Mersenne prime not containing the digit ‘0’ in its decimal expansion.
**Its full name in English is "two billion/milliard one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seven," where the short scale uses "billion" and the long scale uses "milliard."
**DeepLineMadom calls this number TNT as a part of Flenary of Ukrainian Town.Pointless Googolplex Stuffs - Flenary of Ukrainian Town (retrieved 23 June 2022)
*The 9th Mersenne prime is 261-1 = 2,305,843,009,213,693,951.
*The 10th Mersenne prime is 289-1 = 618970019642690137449562111.
*The 11th Mersenne prime \(2^{107}-1\) is the largest known Mersenne prime not containing the digit '4'. Its full decimal expansion is 162,259,276,829,213,363,391,578,010,288,127.
*The 12th Mersenne prime is 2127−1 = 170,141,183,460,469,231,731,687,303,715,884,105,727. It is also a Catalan-Mersenne number.
*The 13th Mersenne prime is \(2^{521}-1 \approx 6.8647976601306097 \times 10^{156}\). It is also a Woodall primeWolfram MathWorld. Woodall PrimeWoodall primes n^2n-1 in the form of \(n \cdot 2^n-1\), where n=512. Its full decimal expansion is 
*The 48th Mersenne prime \(2^{57885161}-1\) was found in Jan 25, 2013 by Curtis Cooper with . It is the largest Mersenne prime where the ranking is determined at September 2024.
*The 49th known Mersenne prime \(2^{74207281}-1\) was found in Jan 07, 2016 by Curtis Cooper with GIMPS. It is not known if there is a Mersenne prime between the 48th Mersenne prime and this number as of September 2024.
*The 50th known Mersenne prime \(2^{77232917}-1\) was found in Dec. 26, 2017 by Jonathan Pace with GIMPS.
*The 51st known Mersenne prime \(2^{82589933}-1\) was found in Dec. 7, 2018 by Patrick Laroche with GIMPS.
== Sources ==
== See also ==
*Largest known prime
== External links ==
* Wikipedia article on Mersenne numbers
* The Great Internet Mersenne Prime Search
* List of known Mersenne primes
*Double Mersenne Prime Search - a distributed computing project to search for divisors of double Mersenne numbers
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