How close can one come to proving that there are infinitely many primes, $p$ and $q$, such that $q = 4p + 1$?
The idea for this question came from reading the question and answers posed by user39898, Quadratic reciprocity and proving a number is a primitive root, URL (version: 2012-12-12): Quadratic reciprocity and proving a number is a primitive root
Based on that result if there are infinitely many pairs of such primes then $2$ is a primitive root modulo infinity many primes.
This would relate to Artin's conjecture on primitive roots: https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots
Because that conjecture has not been proven, all I am looking for is how close we are to proving that such a set of primes is infinite or suggestions on how to proceed. I don't know how to begin answering it.
The tuple $(x, 4x+1)$ is an admissible $2$-tuple in the language of the $k$-tuples conjecture of the conjectures of Hardy and Littlewood. It is believed that there are infinitely many $x$ such that $x$ and $4x+1$ are simultaneously prime. More generally, given an admissible $k$-tuple, it is beleived that there are infinitely many $x$ such that all $k$ elements of the $k$-tuple are simultaneously prime.
We do not know how to prove this. And before a few years ago, there was essentially no major progress towards the $k$-tuple conjecture.
The recent works of Zhang-Maynard-Tao-Polymath8 have made major progress towards a weakened version of the $k$-tuple conjecture. Namely, for any $k$, it has been shown that there exists some $n = n(k)$ (meaning that it depends on $k$) such that for any admissible $n$-tuple, there are infinitely many $x$ making $k$ elements of the $n$-tuple simultaneously prime.
For $2$-tuples, the number is on the order of $250$. What this means is that if one considers a $250$-tuple (like $(x, 4x+1, \ldots, \text{lots of other pieces})$) then there are infinitely many $x$ such that one of the pairs are simultaneously prime. Under the most ideal circumstances, one might hope to reduce the number $250$ to something smaller, but the current methods of proof are not capable of reducing this number all the way down to $2$. Similarly, one might hope to isolate a particular pair, but there is no known way towards this either.
So this is the current state of the art, and there is no current widely-known method or idea of significantly improving this. It should take a genuine new idea (or more likely several new ideas) for progress to be made. I would not be surprised if it took many decades for further significant progress.