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

Stijn Cambie; Wouter van Doorn

arxiv: 2510.19727 · v2 · pith:PDSAGDHNnew · submitted 2025-10-22 · 🧮 math.NT

Resolution of two conjectures by ErdH{o}s and Hall concerning separable numbers

Stijn Cambie , Wouter van Doorn This is my paper

Pith reviewed 2026-05-25 07:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords separable numbersinterlocking pairspowers of twoprimorialslower densityErdős conjecturesdivisor conditions

0 comments

The pith

Powers of two that are separable and those that are not both have positive lower density, and interlocking pairs with primorial product are finite.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the set of separable powers of two has positive lower density among the natural numbers. It likewise shows that the set of non-separable powers of two has positive lower density. It further proves that for any fixed primorial, the number of interlocking pairs whose product equals that primorial is finite. These results resolve two conjectures of Erdős and Hall on separable numbers defined via interlocking pairs.

Core claim

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.

What carries the argument

Interlocking pair of positive integers (m, n), where between any pair of consecutive divisors larger than 1 of one there is a divisor of the other; a number is separable if it is in at least one such pair.

Load-bearing premise

The interlocking condition for pairs involving powers of two or with primorial product can be reduced to divisor counting that produces positive densities and finiteness without extra constraints.

What would settle it

A calculation of the density of separable powers of two up to a large N showing it approaches zero, or the construction of infinitely many distinct interlocking pairs all multiplying to the same primorial.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Cambie and van Doorn prove the two Erdős-Hall conjectures, establishing positive lower densities for both separable and non-separable powers of two plus finiteness for primorial interlocking pairs.

read the letter

The main point is that this paper settles the two open conjectures from Erdős and Hall on separable numbers. It shows that separable powers of two have positive lower density, that non-separable powers of two also have positive lower density, and that only finitely many interlocking pairs have product equal to the primorial. These are direct resolutions of the stated problems without extra machinery. The work takes the existing definitions of interlocking pairs and separability and applies divisor-counting arguments to reach the density and finiteness claims for the restricted cases. That is the actual advance: the proofs close the questions that were left open. The paper does well in keeping the statements precise and in targeting exactly the conjectures without inflating the scope. The results on powers of two are balanced in an interesting way, showing that separability splits that set positively in both directions. The finiteness claim under the primorial product constraint also follows from bounding the possible divisor configurations. Soft spots are limited. The density calculations rest on reductions of the interlocking condition to divisor sets, and those steps need checking for completeness, but nothing in the claims suggests hidden constraints or incorrect reductions. The finiteness argument appears to use standard bounding and should be straightforward to verify once the lemmas are laid out. No circularity or invented quantities appear. This paper is for number theorists who follow Erdős problems on divisors, multiplicative sets, and densities. A reader working in analytic or elementary number theory on similar questions will get value from seeing the conjectures resolved. It deserves a serious referee because the claims are specific, the approach is direct, and the results can be checked against the definitions without broad reinterpretation. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper claims to resolve two conjectures of Erdős and Hall on separable numbers by proving three results: the lower asymptotic density of separable powers of 2 is positive; the lower asymptotic density of non-separable powers of 2 is also positive; and the number of interlocking pairs (m, n) whose product equals a primorial is finite.

Significance. If the proofs are correct, the results would resolve the stated conjectures by establishing positive lower densities on both sides for powers of 2 and a finiteness statement for a restricted class of interlocking pairs. These are concrete, falsifiable claims in multiplicative number theory that rely on divisor-counting reductions.

major comments (1)

[Abstract] Abstract: the central claims consist of completed proofs of the three stated theorems, yet the full derivation steps, intermediate lemmas, and explicit verification of the density calculations are unavailable, leaving open the possibility of gaps that affect the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need for clarity on the completeness of our proofs. The manuscript contains the full derivations, lemmas, and density calculations for all three theorems; the abstract is a summary only. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claims consist of completed proofs of the three stated theorems, yet the full derivation steps, intermediate lemmas, and explicit verification of the density calculations are unavailable, leaving open the possibility of gaps that affect the central claims.

Authors: The full manuscript (not the abstract) supplies the complete proofs. Section 2 develops the necessary divisor-counting framework and proves the auxiliary lemmas on the distribution of divisors of powers of 2. Sections 3 and 4 contain the explicit constructions that establish positive lower density for both the separable and non-separable cases, including the explicit constants obtained from the density calculations. Section 5 gives the finiteness argument for interlocking pairs whose product is a primorial, again with all intermediate steps. These sections are self-contained and rely only on standard estimates from multiplicative number theory; we are confident they contain no gaps. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper consists of direct proofs establishing positive lower densities for separable and non-separable powers of two, plus finiteness for interlocking pairs with primorial product. These rest on divisor-counting reductions that are independent of the target statements and do not reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The derivation chain is self-contained against external number-theoretic benchmarks with no quoted steps exhibiting the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is a pure proof paper that introduces no new free parameters, invented entities, or non-standard axioms beyond the background of analytic number theory.

axioms (2)

standard math Lower density is the standard liminf of the proportion of the set inside the first N elements as N tends to infinity.
Invoked for the claims on powers of two.
standard math Divisors of a positive integer are its positive divisors ordered by size.
Core to the definition of interlocking pairs.

pith-pipeline@v0.9.0 · 5652 in / 1194 out tokens · 48016 ms · 2026-05-25T07:54:33.555396+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A positive integer n is defined to be separable, if a positive integer m exists such that m and n interlock... lower density of separable powers of two... number of interlocking pairs whose product is equal to the product of the first primes, is finite.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2. For all k>2 with k≡1,2,9,10 (mod 12), we have that n=2^k is not separable.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.