---
tag: 0GJBJ19NXR
kind: definition
name: Mersenne number
msc: [11A41]
defines: 0YAG43TCXQ
sources:
  - url: https://en.wikipedia.org/wiki/Mersenne_prime
    locator: About Mersenne primes
---

## Definition

For a natural number $n$, the $n$-th Mersenne number is $M_n = 2^n - 1$.

## Remarks

The first values are $M_1 = 1$, $M_2 = 3$, $M_3 = 7$, $M_4 = 15$ and $M_5 = 31$.
In binary, $M_n$ is written with $n$ digits $1$.
Some texts call $M_n$ a Mersenne number only when $n$ is prime.
A Mersenne number that is prime is a [[Mersenne prime|06GPJF9V7M]].

## Explanations

### L1 human

A Mersenne number is one less than a power of $2$.
For example, $2^{10} - 1 = 1023$, which is $1111111111$ in binary.