Vinogradov's three primes theorem with almost twin primes

TL;DR

This paper extends Vinogradov's three primes theorem by showing that large odd integers can be expressed as sums of three primes with dense prime neighborhoods or near-prime conditions, involving almost twin primes.

Contribution

It proves new variants of Vinogradov's theorem involving primes with dense neighborhoods or almost twin prime conditions for large integers.

Findings

Any large odd integer can be expressed as a sum of three primes with dense prime neighborhoods.

Large integers congruent to 3 mod 6 can be expressed as sums of three primes where each p+2 has at most two prime factors.

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 $N \equiv 3 \pmod{6}$ 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.