The Dirichlet theorem of primes in arithmetic progression tells us that for a given $a$ and $b$ that are coprime there are infinitely integers $k$ s.t $ ak+b $ is prime.
Suppose we take $a=n$, $n$ is odd, and $b=1$ .
For each $n$ and $k$ is the smallest one which makes $nk+1$ prime, what is the best upper bound which we know for $k$ i.e $ k<C $ where C can be a constant or a function depends on $n$ or $k$?