On some additive problems in number theory (v.3)

TL;DR

This paper introduces a new sieve-based method for solving additive problems involving primes, such as the Goldbach-Euler and twin primes conjectures, providing a rigorous approach without heuristic assumptions.

Contribution

It presents a novel Eratosthenes-type double sieve method applicable to prime additive problems, avoiding empirical reasoning and enabling analysis of sequence densities.

Findings

Method applicable to Goldbach-Euler conjecture

Method applicable to twin primes conjecture

Allows analysis of densities of prime sequences

Abstract

In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some interesting results related to the densities of sequences. The method is based on the direct construction of the Eratosthenes-type double sieve and does not use empirical and heuristic reasoning.