Proth Primes: Coefficient & Exponent Relations

[pedja]

Definition: Proth number is a number of the form :

$k\cdot 2^n+1$

where $k$ is an odd positive integer and $n$ is a positive integer such that : $2^n>k$

My question : If Proth number is prime number are there some other known relations in addition to $2^n>k$ , between exponent $n$ and coefficient $k$ ?

 

[pedja]

$( n \equiv 1 \pmod 2 \land n > 1) \Rightarrow \gcd(k-1,3)=1$

$n \equiv 0 \pmod 2 \Rightarrow \gcd(k+1,3)=1$

 

[clearwater304]

I would think it has something to do with the sieve of eratosthenes, where the twin primes revolve around multiples of 6. The formula would give the lesser value for the twin primes.

A few years ago, i decided to look into the riemann hypothesis. I noticed that using the sieve of eratosthenes, there is an obvious pattern for composite numbers. The pattern gets more complex after regions of primes squared. I started to develop a formula but it got more complex with each region, and didn't seam like a good basis for an equation, so I put it off.