On a conjecture of Erd\H{o}s concerning primitive sequences

TL;DR

This paper explores a conjecture about the convergence of series involving primitive sequences and demonstrates through a counterexample that an Erdős-related analogue of this conjecture does not hold.

Contribution

It proposes a new conjecture on series convergence involving primitive sequences and provides a counterexample to an Erdős-related analogue.

Findings

Counterexample disproves the Erdős analogue

Proposes a new conjecture on series convergence

Highlights limitations of Erdős-type conjectures

Abstract

In this note, we propose a conjecture stating that some series involving primitive sequences are convergent. Then, we show (by a counterexample) that the analogue of a conjecture of Erd\H{o}s, for those series, is false.