The IsMersenne(n) command returns $\mathrm{true}$ if $2^n-1$ is prime, $\mathrm{false}$ if $2^n-1$ is not prime, and $\mathrm{FAIL}$ if the status is unknown.

Cases where the status is unknown will generally take an extremely long time to return an answer.

If $2^n-1$ is prime, then it is said to be a Mersenne prime.

There are currently 51 known Mersenne primes.

For every composite number $m$, $2^m-1$ is a not a prime.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{IsMersenne}\left(2\right)$

$\mathrm{true}$

$\mathrm{IsMersenne}\left(3^{36458}\right)$

$\mathrm{false}$

$\mathrm{isprime}\left(60298361\right)$

$\mathrm{true}$

$\mathrm{IsMersenne}\left(60298361\right)$

$\mathrm{FAIL}$

$\mathrm{IsMersenne}\left(60298362\right)$

$\mathrm{false}$

$V≔\left(r\,c\right)\mapsto \mathrm{DocumentTools}:-\mathrm{Tabulate}\left(\mathrm{Matrix}\left(r\,c\,\left(i\,j\right)\mapsto c\cdot \left(i-1\right)+j\right)\,\mathrm{fillcolor}=\left(\left(T\,i\,j\right)\mapsto \mathrm{`if`}\left(\mathrm{IsMersenne}\left(c\cdot \left(i-1\right)+j\right)\,''Pink''\,''White''\right)\right)\right)\:$

$V\left(10\,10\right)$

$''Tabulate''$

The NumberTheory[IsMersenne] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.