In general, can we prove that admissible quadratic polynomials must yield at least one prime value? By admissible, I'm referring to an integer-valued quadratic function with no common factors, i.e. one that meets the basic requirements to be able to produce primes. To be more specific, a quadratic polynomial $f(x):=ax^2+bx+c$ with $a>0$ where  $x\in\mathbb Z \implies f(x) \in\mathbb Z$, and where there exist $m,n\in\mathbb Z$ with $\gcd(f(m),f(n))=1$ and $ m \neq n$. In short, a quadratic which has no a priori reason to be prime-free.
First, I know the infinitude of quadratic primes is an open question, so I suspect this will be too, but I'll ask anyway. Do we know of any quadratic functions that should be able to produce primes yet they don't seem to? By this, I mean an admissible quadratic as described above but for which computer searches have been unable to find any prime values.
I'm assuming not, in which case my primary question is: have we reached the point yet where we can prove, in general, that an admissible quadratic function as described above must yield at least one prime value? Or is that still out of reach?

Upon further thought, I realized that the answer must be no, we currently can't prove it.
If we could, a proof of the infinitude of polynomial primes immediately follows, since you can always split one into arbitrarily many parts. For example, the first six values of $x^2+1$ are $\{2,5,10,17,26,37\}$. But by substituting in $3x-2,3x-1,3x$ to $x^2+1$, it can be completely split into

*

*$9x^2-12x+5$ yielding $\{2,17,\ldots\}$

*$9x^2-6x+2$ yielding $\{5,26,\ldots\}$

*$9x^2+1$ yielding $\{10,37,\ldots\}$
each of which is itself admissible and contains a prime. To ensure admissibility, you either have to split into $k$ pieces where $k$ is not a residue of the original polynomial, or factor out $k$ if it is a residue, but aside from that, I see no reason why this wouldn't work in general, and since I'm sure I'm not the first person to have noticed this, it means that we can't prove that an admissible quadratic must have at least one prime.
