On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

TL;DR

This paper proves new explicit asymptotic results about representing natural numbers as the sum of a prime and a square-free number co-prime to integers with up to two prime factors.

Contribution

It extends previous work by overcoming limitations and making asymptotic results explicit for square-free numbers co-prime to certain integers.

Findings

Every number > 2 can be expressed as a prime plus a square-free number.

New explicit asymptotic formulas are established.

Results apply to square-free numbers co-prime to integers with up to two prime factors.

Abstract

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove new results on square-free numbers co-prime to any integer with up to two prime factors, which make the expected asymptotic results explicit.