I am asking if this variant of the weak Goldbach Conjecture is already known.


Let $N$ be an odd number. Does there exist prime numbers $p\_1$, $p\_2$ and $p\_3$ such that $p\_1+p\_2-p\_3=N$? Ideally, can we find $p\_1$, $p\_2$ and $p\_3$ so that they are small enough? For example, can we prove that for large enough $N$, we can find such a triplet that all of them are smaller than $N$?



A variant of the Goldbach Conjecture
Yes - the standard proof of Vinogradov's result by means of the circle method gives this result. You just need to examine an integral
$$\int\_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^2 \widehat{f}(-\alpha) e(-\alpha N) d\alpha$$
instead of 
$$\int\_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^3 e(-\alpha N) d\alpha.$$
Here $\widehat{f}(\alpha) = \sum\_n \Lambda(n) e(\alpha n) \eta(n/N)$, where $\eta$ is any weight supported in $\lbrack 0,1\rbrack$.


