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A Mersenne number (named after French monk Marin Mersenne) is a number of the form \(2^n - 1\). It generalizes the Mersenne prime, primes that are one less than a power of two, to accommodate for composite instances. Some authors make the additional requirement that \(n\) must be prime, since all Mersenne primes have to have prime exponents.

As of January 2018, there are 50 known Mersenne primes, with \(2^{77,232,917} - 1\) being the largest.

A double Mersenne number is a Mersenne number whose exponent is a Mersenne prime. As of 2016, 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. Catalan conjectured that they are all prime up to a certain limit. 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[]

  • 127 (one hundred twenty-seven) is a positive integer equal to \(2^{2^3-1}-1\). It is notable in computer science for being the maximum value of an 8-bit signed integer. It is the 4th Mersenne prime.
  • 2,047 is the smallest composite Mersenne number with prime index, in this case, (211−1). The next Mersenne number however, which is 213−1 or 8,191, is prime.
  • \(8,191=2^{13}-1\) is the smallest Mersenne prime which is not an exponent of another Mersenne prime.
    • It is also 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 number 17 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Mersenne number is equal to 217−1 = 131,071 or \(M_{17}\). It is also the 6th Mersenne prime.
  • The number 19 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 219−1 = 524,287 or \(M_{19}\). It is also the 7th Mersenne prime.
  • 2,147,483,647 is a positive integer equal to \(2^{31} - 1 = 2^{2^5 - 1} - 1\). It is notable in computer science for being the maximum value of a 32-bit signed integer, which have the range [-2147483648, 2147483647]. It is also a prime number (conveniently for cryptographers), and so the 8th Mersenne prime. 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."
    • The number 31 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture.
    • DeepLineMadom calls this number TNT as a part of Flenary of Ukrainian Town.[1]
  • 9,007,199,254,740,991 is a positive integer equal to \(2^{53} - 1\). It is notable in computer science for being the largest odd number which can be represented exactly in the double floating-point format (which has a 53-bit significand).
    • Its prime factorization is 6,361 × 69,431 × 20,394,401.
  • The number 61 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 261-1 = 2,305,843,009,213,693,951.
  • \(2^{107}-1\) 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.
  • It has been conjectured that no number larger than 127 holds all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 2127−1 = 170,141,183,460,469,231,731,687,303,715,884,105,727.
  • \(2^{521}-1=512*2^{512}-1 \approx 6.8647976601306097 \times 10^{156}\) is the largest known Mersenne prime which is also a Woodall number.
    • Its full decimal expansion is
      6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151.
    • In the fast-growing hierarchy, it is equal to \(f_2(512)-1\)

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