for which values of the pair of integers $(n,k)$ is $p(n,k) =1+\frac{2^{k}-1}n$ is prime?

Question

let $p(n,k)= 1+\frac{2^{k}-1}{n}$ for a positive integer $n,k$

-for which values of the pair of integers $(n,k)$ : $p(n,k)$ is prime ?

Any help is very welcom .Thank you

Answer 1

Call your sum $S$.

Note first that if $n=1$, $S=2^k$, which is not prime. If $n =2^k-1$, then $S =2$ is prime (note that this holds iff).

Assume now that $n$ is a proper factor of $2^k-1$ ($2^k-1$ is no Mersenne Prime n, in which case $S$ could only be an integer for $n=2^k-1)$.

Now, $2^k-1$ has no factor $2$, so it is a product of odd primes. If $n$ is a proper factor, then $\frac{2^k-1}{n}:=f$ is odd too. But then $S=1+f$ is even, and cannot be prime if it is not $2$ (see above).

So $S$ is prime iff $n=2^k-1$.