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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 $$f_2(512)-1$$

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