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

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