Share at least half the numbers in a nontrivial LCM-closed set a nontrivial divisor?

TL;DR

This paper explores a conjecture linking the divisibility properties of non-zero natural number sets with union-closed set conjectures, providing new results and equivalences in number theory, graph, and lattice contexts.

Contribution

It introduces a novel conjecture relating divisibility in natural number sets to union-closed set conjectures and demonstrates its equivalence to Frankl's conjecture.

Findings

Established the conjecture's equivalence to Frankl's union-closed sets conjecture.

Provided results for specific cases where the conjecture holds.

Presented a dual version involving prime power divisibility in natural number sets.

Abstract

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger than 1. Utilizing a representation of the natural numbers as an order-theoretical ring of prime power sets, this conjecture is shown to be equivalent to Frankl's union-closed sets conjecture. Some results for cases where the conjecture, which also has meaningful interpretations in graph and lattice theory, is known to hold are provided. An equivalent dual version of the conjecture is, that for a finite set of non-zero natural numbers that contains at least two elements and the greatest common divisor of any of its subsets, one of its members has a prime power that is not a prime power of more than half of the members.