An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime

Abstract

Call a prime $t$ a twin member if $t-2$ or $t+2$ is prime, and let $\mathcal{T}$ denote the set of twin members. We study the statement (S): every even $n\ge 6$ can be written as $n=p+t$ with $p$ prime and $t\in \mathcal{T}$. Statement (S) sits between the Goldbach conjecture and Dubner's conjecture (every even $n>4208$ is a sum of two twin members), and we observe that (S) is a single elementary statement implying both the Goldbach conjecture and the twin prime conjecture. We prove an orientation-rigidity theorem: for $n\not\equiv 0(\mathrm{mod}3)$, all twin members $t$ appearing in Goldbach partitions of $n$ (representations $n=p+q$ with $p,q$ prime) have the same orientation. According to $n\mathrm{mod}3$, either every such $t$ has $t+2$ prime, or every such $t$ has $t-2$ prime, with degenerate exceptions confined to the primes $3$ and $5$. We verify (S) exhaustively for all even $n\le 10^{12}$. In the course of the earlier run to $10^{11}$ we also re-derive the $33$ terms $\ge 6$ of the Dubner exception list A007534 and re-verify Dubner's conjecture; those two by-products are established for $n\le 10^{11}$ only, the extension to $10^{12}$ covering (S) alone. We compute the least twin witness $t_{\min }(n)=\min \{t\in \mathcal{T}:n-t\text{ prime}\}$, whose maximum is $14549$ (at $n=571714791706$), below ${\log }^{3}n$, and the density of twin-touching Goldbach partitions (those containing a twin member), which decays like $c(n)/\log n$ and matches the Hardy-Littlewood prediction for the constant $c(n)$: measured median values $4.93$-$5.01$ against a predicted median of $4.97$-$4.98$ (the mean is $8C_2\approx 5.28$, with $C_2$ the twin-prime constant).

9 pages. Code and data: doi:10.5281/zenodo.21744861