# 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 $195$.