Cullen number

This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers

Definition

Suppose $n$ is a natural number. The Cullen number parametrized by $n$, denoted $C_{n}$, is defined as:

$C_{n}:=n\cdot 2^{n}+1$

A Cullen number that is a prime is termed a Cullen prime.

Relation with other properties

Weaker properties

• Proth number: A Proth number is a number of the form $k\cdot 2^{n}+1$, $k<2^{n}$. Since $n<2^{n}$ for all $n$, any Cullen number is a Proth number.
• Generalized Cullen number: A Cullen number of the form $n\cdot b^{n}+1$ where $n+2>b$. Note that the special case where $b=2$ gives Cullen numbers.

Other related properties

• Fermat number is a number of the form $2^{2^{n}}+1$, and is also a special case of a Proth number.
• Sierpinski number is a number of the form $k\cdot 2^{n}-1$, with $k<2^{n}$.