Assume Goldbach's conjecture throughout this question. Then every integer $n>1$ admits a non negative integer $r$ such that both $n-r$ and $n+r$ are prime. Let $r_{0}(n)$ the infimum of those integers $r$ and define the fundamental primoradial ratio $R_{n}$ as the ratio $\frac{n+r_{0}(n)}{n-r_{0}(n)}$. I proved in a comment to a former question of mine that $r_{0}(n)<n/2$, so that one has $R_{n}<3$.

Is $R_{n}$ provably maximal for $n=22$? Is it true that $R_{n}<2$ as soon as $n$ is greater than $22$? This would entail $r_{0}(n)<n/3$.

Finally is it true that $\forall\varepsilon>0,\exists m, n>m\Longrightarrow R_{n}<1+\varepsilon$? This would entail $r_{0}(n)=o(n)$.

Edit August 5th 2022: could we prove that $r_{0}(n)\lesssim x_{n}$ where $x_{n}$ is the positive solution of the following equation in $x$:

$x=\frac{n+x}{\log (n+x)}$

which seems to hold numerically, we would get $r_{0}(n)=o(n)$ and $\lim_{n\to\infty}R_{n}=1$. This is inpired by a theorem of Selberg on RH stated in https://www.sciencedirect.com/science/article/pii/0022314X80900311

Moreover, setting $\alpha_{0}:=\inf\{\alpha\mid r_{0}(n)=O(n^{\alpha})\}$, can we get a non trivial upper bound for $\alpha_{0}$, for example assuming GRH?



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