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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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Montgomery and Vaughan showed that the exceptional set in Goldbach's conjecture contains at most $O(x^{1-\delta})$ elements for some $\delta >0$. See . – Lucia Jul 15 '14 at 23:09
What does "cte" mean? – Gerry Myerson Jul 16 '14 at 0:19
@GerryMyerson: "cte" means "constant". I'm sorry, the standard abbreviation is rather "cst"... – Sebastien Palcoux Jul 16 '14 at 0:20
$\text{const}$ or $O(1)$ would have required less head-scratching for me. – Ben Barber Jul 16 '14 at 11:18
I just read your application. I believe it follows more directly, namely a variant of Vinogradov's proof for the ternary Goldbach problem should yield that: any odd number $p$ can be written as $p'+q'-q$ with three primes $p',q',q$. In general, solving a linear equation in three prime variables avoiding the obvious obstructions (e.g. modulo $2$) can be done by the circle method. There is no need to switch to a binary problem. – GH from MO Jul 16 '14 at 17:12
up vote 11 down vote accepted

The weak asymptotic Goldbach conjecture was proved by Chudakov in 1937 (based on the groundbreaking work of Vinogradov). Better bounds are known, see Lucia's comment. Pintz announced that the exceptional set up to $x$ has cardinality $O(x^{2/3})$, but he hasn't published that result yet.

You can find the original reference and a modern treatment in Vaughan's book "The Hardy-Littlewood method".

Added. My response and Lucia's comment hold verbatim for $τ_2$. In fact Montgomery-Vaughan proved that for all but $O(x^{1−δ})$ even integers $x/2<2n<x$, the number of representations $2n=p+q$ is at least $x^{1−3δ}$. Here $δ$ is any sufficiently small positive number, and the implied constant depends only on $δ$. See Section $8$ in the Montgomery-Vaughan paper.

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Wonderful, thank you ! – Sebastien Palcoux Jul 15 '14 at 23:14
@SébastienPalcoux: I mispelled Chudakov's name and year. I will add references in a moment. – GH from MO Jul 15 '14 at 23:14
Let $τ_2(x)$ be the number of even numbers $2<2n<x$ that can't be written as a sum of two distinct pairs of primes. Do you know if we have similar bounds for $τ$ and $τ_2$ ? Perhaps this question requires to open another post, what do you think? – Sebastien Palcoux Jul 15 '14 at 23:56
@Sébastien: The mentioned results not only prove that most even numbers can be written as a sum of two primes, but also that they can be written as a sum of two primes in many ways. Look up Vaughan's book and the Montgomery-Vaughan paper (referred in my response and Lucia's comment). – GH from MO Jul 16 '14 at 0:08
Ok, thank you! Do you know if these results allow to keep similar bounds for $\tau_2$ (see the application I've edited) ? I will look the paper more closely. – Sebastien Palcoux Jul 16 '14 at 0:17

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