Definition:Mersenne Number

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Definition

A Mersenne number is a natural number of the form $2^p - 1$, where $p$ is prime.


The number $2^p - 1$, in this context, can be denoted $M_p$.


Sequence of Mersenne Numbers

The sequence of Mersenne numbers begins:

$3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$

This sequence is A001348 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also defined as

A Mersenne number can also be seen defined as a natural number of the form $2^n - 1$, where $n \in \Z_{\ge 0}$ or $n \in \Z_{> 0}$, but that leads to the singularly boring sequence:

$\paren {0,} 1, 3, 7, 15, 31, 63, 127, 255, 511, 1 \, 023, 2 \, 047, 4 \, 095, 8 \, 191, 16 \, 383, 32 \, 767, \ldots$

This sequence is A000225 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also see

Thus any factors of Mersenne numbers can conveniently be referred to by the value of $k$.


Source of Name

This entry was named for Marin Mersenne.


Sources

where Mersenne number is defined as $2^n - 1$ for all $n \in \N$.
where Mersenne number is defined as $2^n - 1$ for all $n \in \N$.
except he calls it a Mersenne prime by mistake.