A variant of the Goldbach Conjecture 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: Harald Helfgott's proof of the weak Goldbach conjecture can be adapted to prove this result.
A: 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 $\lbrack 0,1\rbrack$.
A: Harald's answer is perfect, but let me tell that using the circle method one can also prove similar results where $p_3$ runs through rather sparse sets (including sparse sets of primes). This is because exceptions to the Goldbach conjecture are relatively rare, i.e. $n+p_3$ will be a sum of two primes (less than $n$) for some odd $p_3<n$, even when $p_3$ is allowed to skip most values. See the works of van der Corput (1937), Tchudakoff (1938), Estermann (1938), Vaughan (1972), Montgomery-Vaughan (1975), etc.
