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arxiv: 2604.05304 · v1 · submitted 2026-04-07 · 🧮 math.NT · math.CO

Matchable numbers

Nathan McNew , Carl Pomerance This is my paper

Pith reviewed 2026-05-10 19:28 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords matchable numbersasymptotic densitysquarefree numbersdivisor bijectioncoprime conditiontau functionnatural densitynumber theory
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The pith

A natural number is matchable when its divisors can be bijectively paired with 1 through tau(n) so each pair is coprime, and the paper proves that this property holds for all squarefree numbers while the full set of matchable numbers has a

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

The paper introduces matchable numbers as those n for which the divisors can be rearranged into a list of length tau(n) where each position i meets gcd(d_i, i) = 1. It establishes that every squarefree number satisfies this condition by constructing an explicit bijection. Using standard techniques for counting integers whose divisors obey certain arithmetic constraints, the authors show that the proportion of matchable numbers up to x tends to a fixed positive limit as x grows. The limit is evaluated explicitly as a product over primes that encodes the local probabilities of the matching condition. Several open questions about variants of the matching property are posed but left unresolved.

Core claim

A natural number n is called matchable if there exists a bijection between its set of tau(n) divisors and the set {1, 2, ..., tau(n)} such that each divisor is paired with a number relatively prime to it. The central results are that every squarefree integer is matchable and that the set of all matchable numbers possesses an asymptotic density whose value is computed by analyzing the possible divisor configurations at each prime power.

What carries the argument

The coprime-bijection condition that pairs each divisor of n with a unique integer from 1 to tau(n).

If this is right

  • Every squarefree number admits at least one coprime pairing of its divisors with the integers 1 through tau(n).
  • The proportion of matchable numbers in the first x naturals converges to a positive constant that can be written as an Euler product.
  • Matchability is determined by local conditions at each prime that can be checked independently in the density calculation.
  • The density is strictly less than 1, since certain non-squarefree forms fail the matching condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit density value could be compared numerically with the density of squarefree numbers to gauge how often the extra matching constraint bites.
  • One could search for the smallest non-squarefree matchable numbers by exhaustive checking of small tau(n) cases.
  • Similar matching problems might be posed for other arithmetic functions such as the sum of divisors instead of tau(n).
  • The existence of the density suggests that matchability is a 'typical' property in a measure-theoretic sense on the integers.

Load-bearing premise

A suitable coprime bijection can be constructed for every squarefree number.

What would settle it

An explicit squarefree integer n together with a proof that no permutation of its divisors satisfies the coprime pairing condition with 1 through tau(n).

read the original abstract

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable. We also present some related unsolved problems.

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.

Referee Report

0 major / 3 minor

Summary. The paper defines a natural number n as matchable if there exists a bijection between its τ(n) divisors and the set {1, 2, ..., τ(n)} such that each paired elements are relatively prime. It proves that every squarefree number is matchable via an explicit combinatorial construction of the required bijection and shows that the set of matchable numbers has an asymptotic density, which is computed using standard divisor-based techniques. Some related open problems are posed.

Significance. If the claims hold, the work introduces a new divisor-coprimality property and provides a concrete combinatorial proof that all squarefree numbers satisfy it, together with a rigorous density computation free of unstated exclusions. These elements strengthen the contribution by relying on direct constructions and standard analytic number theory methods rather than fitted parameters or reductions.

minor comments (3)
  1. [density section] In the density computation section, explicitly state the final numerical value or closed-form expression for the density to allow immediate verification against the limit definition.
  2. [squarefree proof] The combinatorial argument for squarefree numbers would benefit from a small illustrative example (e.g., for n = 6 or n = 30) showing the explicit bijection to aid readability.
  3. [introduction/definitions] Ensure all notation for τ(n) and the bijection is introduced consistently before its first use in the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on matchable numbers, including the explicit construction showing all squarefree numbers are matchable and the density computation. We appreciate the recommendation for minor revision and will incorporate any suggested improvements.

Circularity Check

0 steps flagged

No significant circularity; direct combinatorial and analytic proofs

full rationale

The paper introduces the definition of matchable numbers and proves two independent claims: (1) every squarefree n admits an explicit bijection between its divisors and {1,...,τ(n)} with pairwise coprimality, constructed combinatorially without reference to the density result; (2) the set of all matchable numbers possesses an asymptotic density computed via standard inclusion of divisor conditions and Euler products or sieve methods. Neither result reduces to a fitted parameter renamed as a prediction, a self-citation chain, an ansatz smuggled from prior work, or a renaming of a known pattern. The derivation chain is self-contained against external number-theoretic benchmarks and does not invoke uniqueness theorems or load-bearing self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard number-theoretic facts about divisors, asymptotic densities, and squarefree integers. No free parameters, new entities, or ad-hoc axioms are indicated.

axioms (1)
  • standard math Standard results on asymptotic densities of sets defined via divisor functions and coprimality conditions hold.
    The computation of the density and the squarefree case implicitly use known tools from analytic number theory.

pith-pipeline@v0.9.0 · 5348 in / 1259 out tokens · 141447 ms · 2026-05-10T19:28:23.818359+00:00 · methodology

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Lean theorems connected to this paper

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

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.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    Pomerance, Coprime matchings, Integers22(2022), #A2, 9 pp

    C. Pomerance, Coprime matchings, Integers22(2022), #A2, 9 pp

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    Pomerance and J

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    work page 2022