Mersenne number

This article describes a sequence of natural numbers. The parameter for the sequence is a positive integer (or sometimes, nonnegative integer).
View other one-parameter sequences

Definition

Let $n$ be a natural number. The $n^{th}$ Mersenne number, denoted $M_{n}$, is defined as:

$M_{n}=2^{n}-1$.

Sometimes the term Mersenne number is restricted to the case where $n$ itself is a prime number.

If $M_{n}$ itself is prime, it is termed a Mersenne prime. If $M_{n}$ is prime, so is $n$.

Relation with other properties

Stronger properties

• Sierpinski number

Facts

Number theory facts

• Mersenne number for prime or Poulet implies prime or Poulet
• Mersenne number is prime implies number is prime

Facts in other branches of mathematics

• Order is product of Mersenne prime and one more implies normal Sylow subgroup (fact about groups)

Testing/listing

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A000225

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A001348