# 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 $18$).

The same reasoning can be applied for $n>26$ even and coprime with $3$ by replacing $84$ with $189$ and $90$ with $195$.