Proof that eta(n)>2: case n multiple of 3

Suppose $n>26$ is a multiple of $3$ . One can assume without loss of generality that $r_{0}(n)>2$ . Hence $2r_{0}(n)$ is a Goldbach gap greater than $5$ and thus there are at least three different Goldbach gaps, namely $2$ , $4$ and $2r_{0}(n)$ , hence $\eta(n)>2$ .

This and the previous articles of this blog entail that every even integer greater than $3$ is the sum of two primes.

Proof that eta(n)>2: case n coprime with 3

Suppose $n>26$ is coprime with $6$ . Then every potential typical primality radius of $n$ is a multiple of $6$ . But as $84=6\times 14$ is a multiple of $7$ less than $210$ , and $90=6\times 15$ is a multiple of $5$ less than $210$ , it follows that $84$ and $90$ can’t be potential typical primality radii of $n$ since each of them shares with $n$ a residue class mod $p$ for $p\in\{5,7\}$ .
Hence $\eta(n)>2$ (since one has at least $3$ different Goldbach gaps, namely of size $6$ , $12$ or $18$ ).