Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.

*Goldbach's conjecture*: $\tau(x) = 0$

*Asymptotic Goldbach's conjecture*: $\tau(x) = O(1) $

*Weak asymptotic Goldbach's conjecture*: $\tau(x) = \omicron (\frac{x}{ln(x)}) $

**Question**: Is this weak asymptotic Goldbach's conjecture open ?

What's the better estimate known?

**Application**: Given an odd prime number $p$, there are odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$.

As explained here, we need something slightly stronger for this application: $\tau_2(x) = \omicron (x/ln(x)) $, with $τ_2(x)$ the number of even numbers $2<2n<x$ that can't be written as a sum of two distinct pairs of primes. Is it known ?

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