Vinogradov’s three primes theorem with almost twin primes

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 ∈ {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 ≡ 3 (mod 6) can be written as a sum of three primes p 1 , p 2 and p 3 such that, for each i ∈ {1, 2, 3}, p i + 2 has at most two prime factors.

On a Diophantine Inequality with Prime Numbers of a Special Type

“…Bearing in mind Chen's result, one may try to study the arithmetical properties of the set of primes p such that p + 2 ∈ P r for a fixed r ≥ 2 and, in particular, to establish the solvability of diophantine equations or inequalities in such primes. For example, Matomäki and Shao [11], improving author's results from [17] and [18] as well as a result of Matomäki [10], proved that every sufficiently large odd integer N can be represented as a sum ot three primes p 1 , p 2 , p 3 such that p i + 2 ∈ P 2 , i = 1, 2, 3. Other results of this type were found by the author [19], and by Dimitrov and Todorova [6].…”

Diophantine approximation by special primes

“…Up to now many hybrid theorems were proved. One of the best result belongs to K. Matomäki and Shao [8]. They proved that every sufficiently large odd integer n such that n ≡ 3 (mod 6) can be represented as a sum n = p 1 + p 2 + p 3 of primes p 1 , p 2 , p 3 such that…”

On an equation involving fractional powers with one prime and one almost prime variables

“…In 2000 Tolev [12] proved that for every sufficiently large integer N ≡ 3 (mod 6) the equation (1) has a solution in prime numbers p 1 , p 2 , p 3 such that p 1 + 2 ∈ P 2 , p 2 + 2 ∈ P 5 , p 3 + 2 ∈ P 7 . Thereafter this result was improved by Matomäki and Shao [8], who showed that for every sufficiently large integer N ≡ 3 (mod 6) the equation (1) has a solution in prime numbers p 1 , p 2 , p 3 such that p 1 + 2, p 2 + 2, p 3 + 2 ∈ P 2 .…”

“…Suppose that D, ∆ are define by(10) and ξ, δ are specified by (41). Suppose also that λ(d) satisfy (25) and c m are define by(8).X 1− c 4 ∆ − 1 4…”