Primes of the form $n^2+1$ - hard?

This is an incredibly difficult problem.

It is one of Landau's 4 problems which were presented at the 1912 international congress of mathematicians, all of which remains unsolved today nearly 100 years later.

This problem is hard in the sense that it is still unproven. I will provide a set of references, but little conclusive work (as far as I know) has been done on any of them.

This is a conjecture of Hardy; he later generalized it to say: if a, b, c are relatively prime, a is positive, and $(a+b)$ and c are not both even, and $b^2 - 4ac$ is not a perfect square (I know, quite a set of conditions) - then there are infinitely many primes $an^2 + bn + c$.

He does this on pg. 19 of his book.

I should note that it is proved (even in the same book) that there are infinitely many primes of the form $n^2 + m^2$ and $n^2 + m^2 + 1$. (I'm pretty sure).

There is another statement of this conjecture that is earlier - Are there infinitely many primes $p$ such that $p - 1$ is a perfect square? This is a conjecture of Landau, and it amounts to the same thing (but without Hardy's generalization). As far as I know, the greatest work is to show that there are infinitely many numbers $n^2 + 1$ that have at most 2 prime factors, and it's pretty intense.

Finally, there is a far stronger conjecture called the Horn Conjecture or the Bateman Horn Conjecture. It's a sort of generalization of many other conjectures.

This is a sub-problem of the Bunyakovsky conjecture. I have an interactive form of it at The Bouniakowsky Conjecture. Let $f$ be an integer-coefficient irreducible polynomial with degree higher than 2, and let $k=gcd(f(0),f(1))$.

The conjecture: $f(n)/k$ always generates an infinite number of primes.

Some polynomials, like $x^{12}+488669$ seem to only sparsely make prime numbers, but so far no bounds are known for any of these polynomials.