Mersenne number

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A Mersenne number (named after French monk Marin Mersenne) is a number of the form \(M_n = 2^n - 1\).

When \(M_n\) is prime, it is called Mersenne prime. In order for \(M_n\) to be prime, n must itself be prime. As of September 2024, there are 51 known Mersenne primes, with \(M_{82589933} = 2^{82589933} - 1\) being the largest.

A double Mersenne number is a Mersenne number whose exponent is a Mersenne prime. 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 Eugène Charles Catalan) is a number in the following sequence: 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.
• 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 strong pseudoprime to base 2.
• 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 Goormaghtigh conjecture states that 31 and 8,191 are the only two numbers with this property.
• 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.
• 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 prime in the form of \(n \cdot 2^n-1\), where n=512. Its full decimal expansion is
  6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151.
• The 20th Mersenne prime \(2^{4423}-1\) was found on Nov 3, 1961 by Alexander Hurwitz. It was the first titanic prime ever.
• The 35th Mersenne prime \(2^{1398269}-1\) was found on Nov 13, 1996 by Joel Armengaud with GIMPS. It was the first Mersenne prime to be discovered by GIMPS.
• The 38th Mersenne prime \(2^{6972593}-1\) was found on Jun 1, 1999 by Nayan Hajratwala and Scott Kurowski with GIMPS. It was the first megaprime.
• The 48th Mersenne prime \(2^{57885161}-1\) was found on Jan 25, 2013 by Curtis Cooper with GIMPS. It is the largest Mersenne prime where the ranking is determined at September 2024.
• The 49th known Mersenne prime \(2^{74207281}-1\) was found on 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 on Dec. 26, 2017 by Jonathan Pace with GIMPS.
• The 51st known Mersenne prime \(2^{82589933}-1\) was found on Dec. 7, 2018 by Patrick Laroche with GIMPS.
• The 52nd known Mersenne prime \(2^{136279841}-1\) was found on Oct. 12, 2024 by Luke Durant with GIMPS. It has 41,024,320 digits, which puts it close to a fuganine.

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