How small can the smallest of the three "weak Goldbach" primes always be?

Question

I've checked here for discussions of Helfgott's proof of the weak GC and found nothing that helps me with the following; apologies if I missed something.

I'm probably being naive here (please enlighten me), but I’m thinking that there is a natural conjecture that is intermediate between the strong and weak Goldbach conjectures. The strong GC can be stated as the weak GC (in its "three odd primes" formulation) plus the additional constraint that the smallest of the three primes can always be 3.

My question is: is there a larger number $N$ for which it can already be stated that the smallest of the three primes referred to in the weak GC can always be at most $N$? If not, what can be said about this question?

Answer 1

Your question can be reformulated as: if $n$ is an odd number then how small the odd prime $p$ should be such that $n-p$ is a Goldbach number. Harman, in Chapter 6 of his book Prime-detecting sieves (Princeton University Press, 2007) proved that in the interval $[n-n^{11/180},n]$ all but $O_A(n^{11/180}\log^{-A}n)$ even numbers are Goldbach numbers. It follows (upon taking any $A>1$) that there exists an odd prime $p\ll n^{11/180}$ with the required property. I think this is the state-of-the-art for your question.

Added. Under GRH, much more is known. Kaczorowski-Perelli-Pintz (1993) proved that in the interval $[n-\log^c n,n]$ all but $O(\log^{c/2+3}n)$ even numbers are Goldbach numbers. On the other hand, there are $\gg_c\log^c n/\log\log n$ prime numbers up to $\log^c n$, i.e. more than Goldbach exceptions if $c>6$. It follows that there exists an odd prime $p\ll_\epsilon \log^{6+\epsilon}n$ with the required property.