A conjecture about prime numbers

Question

Is it true that for every integer number $N$ there exist prime numbers $p_1,p_2,p_3,p_4$ such that $N=p_1+p_2-p_3-p_4$ ?

Answer 1

For large enough $N$ this is true. See Vinogradov's theorem.

If $N$ is even, apply the theorem to $N-3$. Otherwise, apply the thorem to $N-2$.