I am sure that this is well-known, but I looked around for the last half hour and couldn't see an answer. I just wondered whether it's possible to insist on taking all primes to be large in Vinogradov type results?


Thanks in advance for any help. 



Is there a constant $c>0$, such that every natural number $n>1$ is the sum of primes, each with size at least $cn$?
The usual proof of Vinogradov's result can be modified to show that every sufficiently large odd $n$ has $\asymp n^2/(\log n)^3$ representations as a sum of three primes with each prime exceeding $cn$, provided $c>0$ is sufficiently small. This gives (easily) a positive answer to your original question.


The best unconditional result of this sort seems to be by Baker and Harman ( R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), 763–780.). They show that every sufficiently large odd $n$ can be written as a sum of three primes from $[\frac{n}{3}-n^{4/7},\frac{n}{3}+n^{4/7}]$.


