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 ≤ 10 5 and even k ≤ 2 • 10 5 that any even integer n ≥ 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.

An Explicit Version of Chen’s Theorem

“…It should be possible to exactly compute k, but we will not do so here. We also note that Theorem 50 was improved by Lee and Francis in [24], and Hathi and Johnston in [27]. However, such improvements have a negligible impact on Theorem 5 unless Theorem 3 is substantially improved.…”

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

“…This is essentially identical to that of [HJ21, Lemma 5.1] with two main differences. First, we have introduced a parameter B which replaces the choice of √ 10 5 used in [HJ21]. Secondly, we have replaced "c θ (da 2 /e)/ log n" with c θ (X3) log 2 (N )…”

“…Note that there is also a slight notation clash with [HJ21, Lemma 5.1]. Namely, N and c θ mean something different in [HJ21] and we have accounted for this accordingly.…”