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 $k<2^n$