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

TL;DR

This paper investigates representations of integers as the sum of a prime and a square-free number under divisibility constraints, providing explicit and asymptotic results and exploring related Goldbach-like problems.

Contribution

It extends previous work by establishing explicit and asymptotic results for sums involving primes and square-free numbers with divisibility conditions.

Findings

Any even integer ≥40 can be expressed as the sum of a prime and a square-free number coprime to k for specified k values.

Provides explicit bounds and asymptotic formulas for such representations under divisibility constraints.

Discusses applications to other Goldbach-like problems.

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 $k\leq 10^5$ and even $k\leq 2\cdot 10^5$ that any even integer $n\geq 40$ can be expressed as the sum of a prime and a squarefree number coprime to $k$. We also discuss applications to other Goldbach-like problems.