Some explicit results on the sum of a prime and an almost prime

TL;DR

This paper proves that every even integer greater than or equal to 4 can be expressed as the sum of a prime and an almost prime with a bounded number of prime factors, with improvements under the generalized Riemann hypothesis.

Contribution

It provides explicit bounds on the number of prime factors in the decomposition of even integers as a prime plus an almost prime, extending classical results.

Findings

Every even integer ≥ 4 is the sum of a prime and a number with at most 395 prime factors.

Under GRH, this bound improves to 31 prime factors.

The results are inspired by and extend classical theorems related to additive number theory.

Abstract

Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 31 prime factors.