The Romanov theorem revised

TL;DR

This paper revises the Romanov theorem by demonstrating that the sumset formed by powers of primes plus a set of products of two primes has a positive lower density, advancing understanding of prime-related sumsets.

Contribution

It proves that the sumset 2^P + P_2 has a positive lower density, providing a new result in the additive properties of primes and their powers.

Findings

The sumset 2^P + P_2 has positive lower density.

This extends previous results on prime sumsets.

The result contributes to additive number theory involving primes.

Abstract

Let P be the set of all primes and P_2=P\cup{p_1p_2: p_1,p_2\in P}$. We prove that the sumset 2^P+P_2={2^p+q: p\in P, q\in P_2} has a positive lower density.