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Resolution of two conjectures by Erdős and Hall concerning separable numbers

Stijn Cambie, Wouter van Doorn

TL;DR

The paper investigates separable numbers and interlocking pairs introduced by Erdős and Hall. It combines divisor-counting arguments, density methods, and prime-gap results to derive new density and finiteness properties. The main contributions are proving that the lower density of separable and non-separable powers of two is positive, identifying congruence classes where $2^k$ is not separable, and proving that interlocking pairs with $mn= P_k$ occur only finitely many times (with $k\le 8$). These results resolve two Erdős–Hall conjectures and deepen understanding of the structure of separable numbers and interlocking pairs.

Abstract

Erdős 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 Erdős and Hall.

Resolution of two conjectures by Erdős and Hall concerning separable numbers

TL;DR

The paper investigates separable numbers and interlocking pairs introduced by Erdős and Hall. It combines divisor-counting arguments, density methods, and prime-gap results to derive new density and finiteness properties. The main contributions are proving that the lower density of separable and non-separable powers of two is positive, identifying congruence classes where is not separable, and proving that interlocking pairs with occur only finitely many times (with ). These results resolve two Erdős–Hall conjectures and deepen understanding of the structure of separable numbers and interlocking pairs.

Abstract

Erdős and Hall defined a pair of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than ) of (resp. ) there is a divisor of (resp. ). 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 Erdős and Hall.
Paper Structure (2 sections, 7 theorems, 5 equations)

This paper contains 2 sections, 7 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Main results and proofs

Key Result

Lemma 1

Let $(m, n)$ be an interlocking pair with $d_2(n) < d_2(m)$. If $n < m$, then $\tau(m) = \tau(n)$, and if $n > m$, then $\tau(m) = \tau(n) - 1$.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 6 more