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arXiv:2202.02384math.NT

A proof of the Erdős primitive set conjecture

Jared Duker Lichtman

Abstract

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum_{a\in A}1/(a \log a)$ 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 Erdős, Sárközy, and Szemerédi from 1968. We also refine the classical Davenport-Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy, and Szemerédi from 1966.

arXiv comments
22 pages. The author was informed that the Erdős primitive set conjecture appears in print at least since 1974, and was possibly posed much earlier still
Journal reference
Forum of Mathematics, Pi (2023)
DOI
doi:10.1017/fmp.2023.16
Subjects
math.NTNumber Theory math.COCombinatorics math.PRProbability

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Linked by 5 identifiable commenters in 5 comments across 2 threads, from Jun 2022 to May 2023.

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Discussed in comments2 threads

  1. Primes and Primitive Sets: an Erdos Conjecture is cracked (2022) [video] 1 comment

    • This appears as the 2nd-last solved conjecture on the List of conjectures by Paul Erdős [1] with 3 references, the first being Jared's paper [2], the 2nd a quanta magazine article [3] and the 3rd this Numberphile video. [1]…

      trompView on HNParent context

  2. Graduate Student’s Side Project Proves Prime Number Conjecture 4 comments

    • Link to Arxiv pre-print: https://arxiv.org/abs/2202.02384

      VicteriusView on HNParent context

    • > Mathematicians noted that the work is particularly striking because it relies entirely on elementary arguments. “It wasn’t like he was waiting for all this crazy machinery to develop,” Thompson said. “He just had some really clever ideas.” the proof…

      nabla9View on HNParent context

    • For anyone who speaks maths https://arxiv.org/abs/2202.02384 Unfortunately without a maths UG it's a hard slog I'll agree, but not the most impenetrable paper over ever seen. Not going to pretend I get the subtleties though.

      rob_cView on HNParent context

    • For anyone interested in a more technical summary of the some of the main ideas: Problem Statement The Erdos conjecture that all primitive sets S satisfy f(S) <= f(PRIMES), where f(S) = sum_{s} (1/(s log s)) and PRIMES is the set of all primes. Definitions Define A_q to be a…

      dragon96View on HNParent context

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