I am sure that this is wellknown, 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.
I am sure that this is wellknown, 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. 


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 $[\frac{n}{3}n^{4/7},\frac{n}{3}+n^{4/7}]$. 

