It is a long-standing conjecture (probably just as old as the twin prime conjecture, which has gotten a lot of attention as of late since Zhang and Maynard's breakthrough results this year) that there exist infinitely many primes of the form $x^2 + 1$. The first few primes of this shape are $5, 17, 37, 101, \cdots$. Of course, no one has been able to prove this theorem, but perhaps if one relaxed the problem one would be able to obtain positive results.
So my question is: what is the least value of $n$ such that one can prove that infinitely many elements of $P_n$ (where $P_n$ is the set of numbers with at most $n$ prime divisors) are of the form $x^2 + 1$?
A similar result to this is a result of Selberg who proved that the polynomial $x(x+2)$ assumes values which have at most 7 prime factors, in particular there exist infinitely many elements of $P_7$ which are of the shape $x(x+2)$.
The parity problem in sieve theory would likely prevent any progress on obtain a lower bound for primes instead of almost primes.
Any insight would be appreciated.