# Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds:

i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an absolute positive constant $C$ such that $g_{n+1}-g_{n}<C$, for all large enough $n$, where $g_{n}$ is the $n$-th Goldbach number,

ii) $\vert\mathcal{D}^{c}\cap[0,N]\vert\le N^{\kappa}$ for some $\kappa<1$ and $N$ a fixed large enough integer, where $\mathcal{D}$ is the set of de Polignac numbers, that is, numbers $m$ such that there exists infinitely many $i$ such that $p_{i+1}-p_{i}=m$, where $p_{i}$ denotes the $i$-th prime number.

Assuming i) holds, is it possible to make $C$ explicit with current technology?

• I think it would be a fantastic result if we could prove that the existence of $C$ implies a concrete value for $C$. This result is not in the literature, so it is fair to say that it is not possible to make $C$ explicit with current technology. Of course any new paper might change this opinion, e.g. Zhang and Maynard-Tao also used existing technologies, but noone believed a few years ago that their results were within reach. – GH from MO May 14 '15 at 20:55
• I read somewhere (wikipedia?) that it has been proven that the set of Goldbach numbers has natural asymptotic density one, so maybe an explicit value for $C$ is attainable. – Sylvain JULIEN May 14 '15 at 21:00
• Density one is an old result, about 75 years old. By more elaborate techniques it is even known that up to $x$ there are only $O(x^{2/3})$ exceptions to Goldbach's conjecture (Pintz, unpublished). Establishing the existence of $C$ or bounding it (assuming its existence) is another matter. But who knows, maybe more breakthroughs are around the corner. – GH from MO May 14 '15 at 21:05