{
  "abstract": "A sequence of integers $1=a_0<a_1<\\cdots<a_k=n$ is called an addition chain of length $k$ if $a_j=a_s+a_t$ with $0\\le s,t<j$ for all integers $j\\in \\{1,2,\\ldots,k\\}$. We denote by $\\ell(n)$ the minimal length of an addition chain leading to $n$. Here we investigate the distribution of the function $\\ell$ through the counting function $$ F(m,r):=\\#\\{n\\in [2^m,2^{m+1}):\\ell(n)\\le m+r\\} $$ and show that, for every fixed $0<c<\\log 2$, there exist positive constants $K_1$ and $K_2$ such that $$ K_1^r m^r\\le F(m,r)\\le K_2^r m^r $$ for all sufficiently large $m$ and all integers $m^{0.9}<r\\le cm/\\log m$. The upper bound also holds for every integer $r>m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of \\emph{distinct} addition chains of length $m+r$ leading to an integer $n\\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \\left(\\frac{m^2}{r}\\right)^r\\le G\\left(m,r\\right)\\le K_4^r \\left(\\frac{m^2}{r} \\right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28374",
  "author_total": 3,
  "authors": [
    {
      "name": "Jean-Marie De Koninck"
    },
    {
      "name": "Nicolas Doyon"
    },
    {
      "name": "William Verreault"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33trtn8xebvq6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28374",
  "fulltext_url": null,
  "id": "33trtn8xebvq6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28374",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1710|Cryptography and Residue Arithmetic"
    ]
  ],
  "openalex": "W7214164119",
  "pdf_url": "https://arxiv.org/pdf/2609.28374",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33trtn8xebvq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28374",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28374"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the distribution of the minimal length of addition chains",
  "title_text": "On the distribution of the minimal length of addition chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/33trtn8xebvq6",
  "venue": "arXiv",
  "versions": []
}