A famous old conjecture is that infinitely many primes of form n^2+1
I have no idea how to settle that, but can one instead settle this easier problem:
are there infinitely many n such that n^2+1 is either prime or "P2"
(product of two primes)?
The answer is YES!
Harald A. Helfgott: On the square-free sieve, Acta Arithmetica 115,4 (2004) 349-402
T. Estermann: Einige S"atze "uber quadratfreie Zahlen, Math. Annalen 105 (1931) 653-662.
we know that there are infinitely many n^2+1 which are squarefree.
Henryk Iwaniec: Almost-primes represented by quadratic polynomials
Inventiones math. 47,2 (1978) 171-188
we know that any polynomial P(n) that is not trivially excluded
represents an infinity of numbers that are products of <=k primes,
where k=degP + const*log(degP).
Finally, Iwaniec as his Main Theorem shows there are an infinite set of n such
that n^2+1 is a product of at most two primes. Also works for 4*n^2+1 and
indeed for any irreducible quadratic P(n) with P(0)=1, and the count of such n
below X is show to grow eventually at last proportionally to X/logX;
proportionality constant depends on P in a simple way.