Vinogradov's three primes theorem with almost twin primes

Abstract

In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any in, every sufficiently large odd integer N can be written as a sum of three primes p(1), p(2) and p(3) such that, for each i is an element of {1, 2, 3}, the interval [p(i), p(i) + H] contains at least m, primes, for some H = H (m). Second, every sufficiently large integer N 3 (mod 6) can be written as a sum of three primes p(1), p(2) and p(3) such that, for each i is an element of {1, 2, 3}, p(i) + 2 has at most two prime factors.