# MersennePrimeExponentQ

returns True if n is a Mersenne prime exponent, and False otherwise.

# Details • MersennePrimeExponentQ is typically used to test whether an integer is a Mersenne prime exponent.
• A positive integer n is a Mersenne prime exponent if the Mersenne number is prime.
• • returns False unless n is manifestly a Mersenne prime exponent.

# Examples

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## Basic Examples(2)

Test whether a number is a Mersenne prime exponent:

The number is not a Mersenne prime exponent:

## Scope(3)

MersennePrimeExponentQ works over integers:

Negative integers are not Mersenne prime exponents:

Noninteger numbers are not Mersenne prime exponents:

## Applications(8)

### Basic Applications(3)

Highlight Mersenne prime exponents:

Generate random Mersenne prime exponents:

Digits of a Mersenne prime :

### Special Sequences(2)

Recognize Mersenne numbers, numbers of the form :

The number is a Mersenne number; is not:

Recognize Gaussian Mersenne primes, prime numbers n such that is a Gaussian prime:

### Number Theory(3)

A trinomial whose order is a Mersenne prime exponent is primitive modulo if and only if it is irreducible:

If p is a Mersenne prime exponent, then is a perfect number:

Every even perfect number has the form , where p is a Mersenne prime exponent:

Check that in the representation above p is 5:

## Properties & Relations(10)

Mersenne prime exponents are prime numbers:

Composite numbers cannot be MersennePrimeExponents:

The only even Mersenne prime exponent is :

MersennePrimeExponent gives  Mersenne prime exponent: is a Mersenne prime, where p is a Mersenne prime exponent:

If p is a Mersenne prime exponent, then is a perfect number:

Every even perfect number has the form , where p is a Mersenne prime exponent:

Check that in the representation above p is 5:

Triangular numbers of Mersenne primes are perfect numbers:

Hexagonal numbers related to Mersenne prime exponents are perfect numbers:

Find Mersenne prime exponents:

## Possible Issues(1)

Expressions that represent Mersenne prime exponents but do not evaluate explicitly will give False:

It is necessary to use symbolic simplification first:

## Neat Examples(1)

The minor planet 8191 Mersenne is named after Marin Mersenne:

The number is a Mersenne prime:

Introduced in 2016
(10.4)