Vinogradov's three primes theorem with almost twin primes. (20th April 2017) - Matomäki, Kaisa; Shao, Xuancheng
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 anym, every sufficiently large odd integerN can be written as a sum of three primesp_1, p_2 andp_3 such that, for eachi{1, 2, 3}, the interval[p_i, p_i+H] contains at leastm primes, for someH=H(m) . Second, every sufficiently large integerN3 (mod 6) can be written as a sum of three primesp_1, p_2 andp_3 such that, for eachi{1, 2, 3}, p_i+2 has at most two prime factors.