TITLE: Tightest proven upper bounds for the smallest prime of the for $an + b$
QUESTION [5 upvotes]: If $\gcd(a, b) = 1$, it is known that there are infinitely many primes of the form $an + b$. 
Denote $n(a, b)$ to be the smallest $n$ such that $an + b$ is prime.
$n(a, b) \leq \max(a, b)$ holds for $1 \leq a, b\leq 10000$.

What is the tightest PROVEN upper bound for $n(a, b)$ ?

REPLY [3 votes]: This is a well-studied problem, a key-word is "Linnik's theorem." 
Let us denote the least prime itself by $p(a,b)$, so $p(a,b) = b + n(a,b)a$. 
It is more common to express results in that way, and one can pass from on to the other easily. 
Then Linnik proved $p(a,b) \le c a^L$ for some constants $c$ and $L$. 
Meanwhile the best constant $L$ for which this is known is $L=5$. 
At least for $L=5.5$ the constant $c$ is effectively computable (I am not sure an actual value was determined though).  
It is conjecture that $L=2$ is true, and more precisely that $p(a,b) < a^2$.
Under the generalized Riemann hypothesis it is known that $p(a,b) \le (1+o(1)) \varphi(a)^2  \log(a)^2$.
For further details see the linked page.