# 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.