{
  "abstract": "We study the Erdős-Moser equation in arithmetic progressions, $\\sum i^k=m^k$, where $i$ runs over the positive integers $i<m$ with $i\\equiv b \\pmod q$. We prove that a solution has at most $\\lfloor k/\\log 2+3/2\\rfloor$ terms for all $k<10^{55}$, and at most $k/\\log 2+5/2$ terms in general; this linear bound is best possible for real solutions and replaces the previously known exponential bound $(k+1)2^{2k+1}$. For $k=3$ we obtain a complete classification: solutions with exactly $n$ terms exist if and only if $n\\in\\{3,4,5\\}$, and in each case there are infinitely many primitive solutions. For even exponents there are no solutions with three terms, and there are no solutions at all for $k=4$, for every $k=2^s\\ge 4$, and for every even $k$ with $4\\le k\\le 24$. The key analytic input is a uniform estimate for the classical equation $1^k+\\dots+(m-1)^k=m^k$: the real exponent $κ_m$ for which $1^κ+\\dots+(m-1)^κ=m^κ$ satisfies $κ_m=\\log 2\\,(m-3/2-γ_m/m)$ with $0<γ_m<1/(8\\log 2)$ for every $m\\ge 3$. The proofs rest on a method based on centres of mass, Steiner's parallel-axis theorem and what we call the half-mass principle. The same framework identifies the constants of a general class of balance equations through the moments of a mass distribution: $3/2$ is a Steiner quotient, and $25/12-3\\log 2=1/12-f''(\\log 2)/2$ with $f(λ)=λ/(e^λ-1)$, whose Bernoulli expansion explains why this constant is so small.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39315",
  "author_total": 2,
  "authors": [
    {
      "name": "Elham Mehdi-Nezhad"
    },
    {
      "name": "Amir M. Rahimi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ydwnzx7jyydd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39315",
  "fulltext_url": null,
  "id": "6ydwnzx7jyydd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39315",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ydwnzx7jyydd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Centres of mass, Steiner's theorem, and equations of Erdős-Moser type",
  "title_text": "Centres of mass, Steiner's theorem, and equations of Erdős-Moser type",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/6ydwnzx7jyydd",
  "venue": "arXiv",
  "versions": []
}