{
  "abstract": "For a positive integer $n$, let $D(n)$ be its set of positive divisors and put $N:=τ(n)=|D(n)|$. We characterize the arithmetic progressions $\\mathcal P$ of length $N$ that admit a bijection with $D(n)$ pairing relatively prime integers. Such a coprime matching exists if and only if, for every prime $p\\mid n$, at most $N/(ν_p(n)+1)$ terms of $\\mathcal P$ are divisible by $p$, where $ν_p(n)$ is the exponent of $p$ in $n$. Call $n$ an $M$-number if $p^p\\nmid n$ for every prime $p$. We prove that $D(n)$ can be matched with every primitive arithmetic progression of length $N$, meaning one whose first term and common difference are coprime, if and only if $n$ is an $M$-number. This proves a conjecture of McNew and Pomerance. We call $n$ \\emph{matchable} if $D(n)$ admits a coprime matching with the primitive arithmetic progression $\\{1,2,\\ldots,N\\}$. The general criterion implies that $n$ is matchable if and only if $\\lfloor N/p\\rfloor\\le N/(ν_p(n)+1)$ for all $p\\mid n$. We explicitly classify the matchable non-$M$-numbers, settle a conjectured divisor criterion, and prove that every divisor of a matchable integer is matchable. A Vandermonde determinant gives the required matchings under bounds on the prime exponents, while a partition into shorter progressions handles the remaining cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37512",
  "author_total": 1,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9f24qw419s6s1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37512",
  "fulltext_url": null,
  "id": "9f24qw419s6s1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37512",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7224078261",
  "pdf_url": "https://arxiv.org/pdf/2610.37512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9f24qw419s6s1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Coprime matchings of divisor sets",
  "title_text": "Coprime matchings of divisor sets",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9f24qw419s6s1",
  "venue": "arXiv",
  "versions": []
}