Vinogradov's three primes theorem with almost twin primes

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 $N\equiv 3(\mathrm{mod}6)$ 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.

41 pages

References in corpus (1)

• The ternary Goldbach problem

Cited by in corpus (11)

• Vinogradov's theorem with almost equal summands
• The Goldbach Problem for Primes That Are Sums of Two Squares Plus One
• Vinogradov's Theorem with Fouvry-Iwaniec Primes
• Diophantine approximation by special primes
• On the Twin Prime Conjecture
• On a diophantine inequality with prime numbers of a special type
• The Bombieri-Vinogradov theorem for nilsequences
• A transference principle for systems of linear equations, and applications to almost twin primes
• Gaps between primes
• On an equation involving fractional powers with prime numbers of a special type
• Products of primes in arithmetic progressions