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.