On the finiteness of some p-divisible sets

Abstract

For any positive integer $n$, let $H_n$ denote the $n^{th}$ harmonic number. Given a prime number $p$, it is not known whether the set of integers $J(p) = \{n \in \mathbb{N} : p \mid H_n \} $ is finite. In this paper, we first investigate a variant of this set, namely, we work on the divisibility properties of the differences of harmonic numbers. For any prime $p$ and a positive integer $w$, we define the set $D(p,w)$ as $\{n \in \mathbb{N} : p \mid H_n - H_w \}$ and work on the structure of this set. We present some finiteness results on $D(p,w)$ and obtain upper bounds for the number of elements in the set. Next, we consider the differences of generalized harmonic numbers and present an upper bound for the corresponding counting function. Moreover, under some plausible conditions, we prove that the difference set of generalized harmonic numbers is finite. Finally, we point out some directions to pursue.