I have received [a complaint](https://mathoverflow.net/questions/80865/least-prime-in-a-arithmetic-progression/80867#comment537038_80867) about my 2011 answer 


[least prime in a arithmetic progression](https://mathoverflow.net/questions/80865/least-prime-in-a-arithmetic-progression/80867#80867) 


which, indeed, gives conflicting reports about this:



> 
> given a prime $q,$ what can we say about an upper bound for the smallest prime $p$ in the arithmetic progression $n q + 1$?
> 
> 
> 


Note that I do not see anything using the number $70$ in [THIS](https://en.wikipedia.org/wiki/Linnik%27s_theorem).


I guess there are about three parts:


(A) What is the most optimistic upper bound, i.e. numerical computations? I had a short computer run in my answer.


(B) What is the strongest result one gets assuming a generalized/extended Riemann Hypothesis?


(C) What is the strongest unconditional result? 



Update for 2015: least prime of form nq+1, with q prime?
The most optimistic conjecture is that the least prime in this (or indeed any progression $a\pmod q$) is $\ll q (\log q)^2$. This is an analog of Cramer's conjecture on primes in short intervals, so way beyond reasonable conjectures like GRH! 


On GRH [Lamzouri, Li, and Soundararajan](http://arxiv.org/pdf/1309.3595v1.pdf) (see Corollary 1.2) have shown that the least prime that is $a\pmod q$ (with $q>3$ and $(a,q)=1$) is bounded by 
$$ 
\le (\phi(q) \log q)^2. 
$$ 
Note this is an explicit inequality. Moreover they note that asymptotically one could get an estimate $\le (1-\delta +o(1)) (\phi(q)\log q)^2$ for some $\delta >0$. 


Unconditionally, Linnik was the first to show that the least prime is $\ll q^{L}$ for some fixed $L$, and over the years this has been improved and the current record is that $L=5$ is permissible due to Xylouris. 


