In trying to work on the question of whether there are infinite primes of form $x^2+1$, there's one issue I really don't get, and was hoping somebody might be good enough to help me out.
Most of my thinking on the issue starts with accepting the assertion that there are only finitely many $x^2+1$ primes, and then poking around the implications of that in hopes of finding something exploitable to disprove. (As an amateur, I know I might as well be playing the lottery, but hey.)
Finitely many $x^2+1$ primes implies there's some maximum $x^2+1$ which is prime. Thanks to Iwaniec, we know we at least get infinitely many semiprimes after that, but my impression is that we can't say much more concretely about their makeup and distribution. I reason that there can only be finitely many semiprimes of a $2p$ form, or $5p$, or actually any $kp$ for fixed integer $k$, since otherwise we could divide through by $k$ to get an infinite-prime polynomial.
I mention the above in case any of that is off track, but my main question is: do we have any concept of how such a covering system could exist, and what it would look like if it did? Since we can't cover infinitely many primes with a finite set, and we can't use any congruence tricks, I have absolutely no feel for this idea in general. It seems like it would require a never-ending stream of perfectly timed $4k+1$ primes falling into place, forever.
To clarify: I am asking for a little explanation of how a situation like this might arise, or maybe some useful keywords to search, or a link to a paper, whatever might help me wrap my head around it. Thanks.