# Definition:Proth Number

## Definition

A Proth number is a natural number of the form:

$N = k \times 2^n + 1$

where:

$k$ is an odd integer
$n$ is a positive integer
$2^n > k$

The condition is included, otherwise every odd integer greater than $1$ would be a Proth number.

### Sequence of Proth Numbers

The sequence of Proth numbers begins:

$3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, \ldots$

## Source of Name

This entry was named for François Proth.