A proof of the ErdÅ s primitive set conjecture

Abstract

A set of integers greater than 1 is primitive if no member in the set divides another. ErdÅ s proved in 1935 that the series is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative. As further applications of the method, we make progress towards a question of ErdÅ s, Sárközy and Szemerédi from 1968. We also refine the classical Davenport-ErdÅ s theorem on infinite divisibility chains, and extend a result of ErdÅ s, Sárközy and Szemerédi from 1966.