Proth number

Definition

Suppose $n$ is a natural number and $k$ is a natural number such that $2^{n}>k$. The Proth number with parameters $n$ and $k$ is defined as the number:

$k\cdot 2^{n}+1$.

A Proth number that is also a prime is termed a Proth prime.

Relation with other properties

Stronger properties

• Fermat number: The $m^{th}$ Fermat number is the Proth number with $k=1$ and $n=2^{m}$.
• Cullen number

Other related properties

• Mersenne number is a number of the form $2^{n}-1$.
• Sierpinski number is a number of the form $k\cdot 2^{n}-1$ with $k<2^{n}$