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

An earlier version of this work contained material in two directions, so a decision was made to split the old paper into two focussed papers that are more reader-friendly. This is one such paper, the other is titled "On the sum of a prime and a number that is not square-free" and will be released separately. Additional info: 14 pages, feedback warmly welcomed as usual