Extension of Goldbach's conjecture to polynomials

I noticed that a slightly modified version of Goldbach's conjecture seems to hold for the quadratic $$x^2+1$$. Specifically, I assert for any even $$n\geq 4$$, there exists at least one pair $$p,q\in\mathbb N$$ such that: $$p^2+1\in\mathbb P\\ q^2+1\in\mathbb P\\ p+q=n .$$

Prime $$x^2+1$$ are fairly plentiful, starting with $$x$$ values $$\{1,2,4,6,10,14,16,\ldots\}$$, so it's not too hard to believe. There are a few small $$n$$ which only have a single $$p,q$$ pair, but soon enough the number of pairs takes off, as in Goldbach proper.

I found other $$x^2+k$$ forms that seem to share this property, including $$k=7, 25,37,43,\ldots$$. Those $$k$$ that start out more prime dense are more likely to make the cut, but there are exceptions, like $$x^2+187$$, which is denser than $$x^2+1$$ but has two early failures.

We can simplify a little by tweaking the polynomials in question. For instance, if we use $$4x^2+1$$ instead of $$x^2+1$$, it removes the even values, so we can get rid of the evenness constraint and claim $$p+q=n$$ for all $$n \geq 2$$.

Further investigation suggested the following:

For any admissible polynomial $$P(x)$$, there exists some $$M$$ such that for all $$n\geq M$$, there is at least one pair $$p+q=n$$ with prime $$P(p)$$ and $$P(q)$$.

In other words, every polynomial that is able to will asymptotically comply with Goldbach. Of course, this presupposes the existence of infinitely many polynomial primes, as with Bateman-Horn or comparable.

By admissible polynomial, I refer to integer-valued polynomials which aren't trivially excluded by pre-existing conditions, like those that are reducible or otherwise fail to meet the basic congruency criteria for infinitude of primes. Beyond that, I specifically exclude two more cases, both of which make Goldbach compliance impossible:

1) As mentioned before, those polynomials where every other term is even.

2) Any polynomial having $$3$$ as a quadratic residue, such that two out of every three terms are divisible by $$3$$.

Apart from those constraints, all indications are that polynomials tend to become Goldbach-compliant inevitably, and in general, pretty quickly. I would add the caveat that this is based off playing with quadratics and to a lesser extent cubics, so I'm only guessing that it applies to higher degrees; my confidence mostly applies to quadratic forms.

Anyway, I've seen a number of generalizations of Goldbach's conjecture but haven't found any that deal with this. I'm curious whether this has been noticed before, and otherwise invite counterexamples, corrections and comments.

• Interesting, what makes this extension even harder is that we do not even known whether $n^2+1$ produces infinite many primes (which is of course expected to be the case, but we have not more evidence than for the truth of Goldbach's conjecture) – Peter Aug 25 at 13:17
• Every even positive integer $n\le 9\cdot 10^6$ is the sum of two positive integers $a$ and $b$ , such that both $a^2+1$ and $b^2+1$ are prime (found out with pari/gp using brutce force) – Peter Aug 25 at 14:53