The Bateman-Horn conjecture is a generalization of the twin prime conjecture that roughly states that given a set $S=\{f_1, \dots, f_m\}$ of irreducible polynomials with integer coefficients, there are infinitely many positive integers $n$ such that $f_1(n), \dots, f_m(n)$ are all prime, provided that there are no obvious congruence conditions on the values of the polynomials that immediately imply otherwise. Additionally, it states that the number of such $n\leq x$ is asymptotically $x/\log(x)^m$ times a constant depending on $S$.
We can formulate a similar generalization of the Goldbach conjecture to arbitrary sets of polynomials: given a set polynomials $S=\{f_1, \dots, f_m\}$, satisfying the conditions of the Bateman-Horn conjecture, there are constants $k$ and $N$ depending on $S$, such that every positive integer greater than $N$ is the sum of at most $k$ positive integers $n$ for which all $f_1(n), \dots, f_m(n)$ are prime, provided that there are no obvious congruence conditions on the values of $n$ that immediately imply otherwise.
Some instances of this conjecture are as follows:
- $S=\{x\}$ with $k=3$ and $N=1$ (the standard Goldbach conjecture)
- $S=\{x,x+2\}$ with $k=5$ and $N=7$
- $S=\{x,x+4\}$ with $k=5$ and $N=644$
- $S=\{x,x+6\}$ with $k=3$ and $N=8006$
- $S=\{x,x+2,x+6\}$ with $k=9$ and $N=29$
- $S=\{x,x+4,x+6\}$ with $k=9$ and $N=382$
- $S=\{x,x+2,x+6,x+8\}$ with $k=12$ and $N=1189613$
- $S=\{x,2x+1\}$ with $k=4$ and $N=7021$
- $S=\{x^2+1\}$ with $k=3$ and $N=0$
Questions:
- Has this conjecture (or something similar) been formulated or written about in the literature before? I was not able to find anything during my search.
- Can we heuristically determine how large asymptotically $k$ and $N$ should be in terms of the polynomials in $S$?
- Can we heuristically determine asymptotically how many representations a large positive integer should have as a sum of such values of $n$?
