Resolution of two conjectures by Erd\H{o}s and Hall concerning separable numbers

TL;DR

This paper proves two conjectures by Erdős and Hall regarding the properties of separable numbers and interlocking pairs, showing positive density results and finiteness of certain interlocking pairs.

Contribution

It resolves two longstanding conjectures by Erdős and Hall about separable numbers and interlocking pairs, establishing density and finiteness results.

Findings

Positive lower density of separable powers of two.

Positive lower density of powers of two that are not separable.

Finiteness of interlocking pairs with product equal to the product of the first primes.

Abstract

Erd\H{o}s and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$) of $n$ (resp. $m$) there is a divisor of $m$ (resp. $n$). A positive integer is said to be separable if it belongs to an interlocking pair. We prove that the lower density of separable powers of two is positive, as well as the lower density of powers of two which are not separable. Finally, we prove that the number of interlocking pairs whose product is equal to the product of the first primes, is finite. We hereby resolve two conjectures by Erd\H{o}s and Hall.