Vinogradov's three primes theorem with almost twin primes. (20th April 2017)

• Record Type: Journal Article
• Authors: Matomäki, Kaisa; Shao, Xuancheng
• Is Part Of: Compositio mathematica. Volume 153:Number 6(2017)
• Page Start: 1220
• Page End: 1256
• Publication Date: 2017-04-20
• Subjects: 11P32, -- 11P55, -- 11N35 (primary); Vinogradov's theorem, -- transference principle, -- sieve method, -- twin primes, -- Bohr sets; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007072
• Languages: English
• ISSNs: 0010-437X

Abstract

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.