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A Mersenne number (named after French monk Marin Mersenne) is a number of the form \(2^n - 1\). Some authors make the additional requirement that n must be prime. A Mersenne prime is a Mersenne number that is prime.

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.
  • 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 \(f_2(512)-1\)

Sources


See also

External links

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