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Vinogradov’s three primes theorem with almost twin primes

Part of: Multiplicative number theory Additive number theory; partitions

Published online by Cambridge University Press:  20 April 2017

Kaisa Matomäki  and
Xuancheng Shao
Kaisa Matomäki
Affiliation:
Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland email [email protected]
Xuancheng Shao
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
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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 $p_{1},p_{2}$ 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.

Keywords

Vinogradov’s theoremtransference principlesieve methodtwin primesBohr sets

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes 11P55: Applications of the Hardy-Littlewood method 11N35: Sieves
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 6 , June 2017 , pp. 1220 - 1256
Copyright
© The Authors 2017 

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