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:

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

Also see

• Riesel Problem
• Definition:Carmichael Number
• Definition:Sierpiński Number of the Second Kind

• Results about Riesel numbers can be found here.

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.

Sources

• 1989: Paulo Ribenboim: The Book of Prime Number Records (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $509,203$

• Weisstein, Eric W. "Riesel Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RieselNumber.html