Primitive sets and von Mangoldt chains: Erd\H{o}s Problem #1196 and beyond

TL;DR

This paper introduces a novel Markov chain-based method for analyzing primitive sets and Erdős sums, successfully resolving several longstanding conjectures and providing new insights into divisibility structures.

Contribution

The authors develop a new approach using von Mangoldt-weighted Markov chains to bound Erdős sums, leading to proofs of multiple Erdős conjectures and a resolution of the Banks-Martin conjecture.

Findings

Proved two Erdős-Sárközy-Szemerédi conjectures from 1966.

Provided a short proof of the Erdős Primitive Set Conjecture.

Resolved the Banks-Martin conjecture.

Abstract

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erd\H{o}s's seminal 1935 paper. As applications, we prove two 1966 conjectures of Erd\H{o}s-S\'ark\"ozy-Szemer\'edi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erd\H{o}s Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erd\H{o}s-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area.