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 any$m$, 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$i\in \{1, 2, 3\}$, the interval$[p_{i}, p_{i}+H]$ contains at least$m$ primes, for some$H=H(m)$ . Second, every sufficiently large integer$N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes$p_{1}, p_{2}$ and$p_{3}$ such that, for each$i\in \{1, 2, 3\}$, $p_{i}+2$ has at most two prime factors.