# Vinogradov's three primes theorem with almost twin primes

Kaisa Matomäki, Xuancheng Shao

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## 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 p1, p2 and p3 such that, for each i ϵ (1, 2, 3), the interval [pi, pi + H] contains at least m primes, for some H = H(m). Second, every sufficiently large integer N Ξ 3 (mod 6) can be written as a sum of three primes p1, p2 and p3 such that, for each i ϵ (1, 2, 3), pi + 2 has at most two prime factors.

Original language English 1220-1256 37 Compositio Mathematica 153 6 https://doi.org/10.1112/S0010437X17007072 Published - Jun 1 2017

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