# Definition:Riesel Number

## Definition

Let $k$ be an odd positive integer.

Then $k$ is a Riesel number if and only if:

For all positive integers $n$, the number $k \, 2^n - 1$ is composite.

### Sequence of Known Riesel Numbers

The sequence of known Riesel numbers starts:

$509 \,203, 762 \,701, 777 \,149, 790 \,841, 992 \,077, 1 \,106 \,681, 1 \,247 \,173, \ldots$

## Source of Name

This entry was named for Hans Ivar Riesel.

## Historical Note

In $1956$, Hans Ivar Riesel proved that there are an infinite number of Riesel numbers.