On the sum of a prime and a square-free number with divisibility conditions

Abstract

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k≤105 and even k≤2⋅105 that any even integer n≥40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss applications to other Goldbach-like problems.