TITLE: Update for 2015: least prime of form nq+1, with q prime?
QUESTION [11 upvotes]: I have received a complaint about my 2011 answer 
least prime in a arithmetic progression 
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.
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?

REPLY [2 votes]: The $70$ comes from the claimed result of Oesterle mentioned here (which is supposed to use GRH only).