{
  "abstract": "We introduce a probabilistic version of the iterated factor method introduced in our previous investigations. Let $n$ be drawn uniformly from $\\{1,2,\\ldots,M\\}$. Set $$ s:=s(n)=\\left \\lfloor \\frac{\\sqrt{\\log_2n}}{\\log \\log_2n} \\right\\rfloor \\quad \\text{and} \\quad t(n)=\\sqrt{\\log \\log n} $$ and $$ k_n=\\left \\lfloor \\frac{n}{2}\\right \\rfloor_s. $$ For $N\\longrightarrow \\infty$ with $Z_n\\sim N(0,1)$~(normally distributed) and additionally that $$ \\mathrm{Pr}[\\nu(k_n)\\leq s \\quad \\text{and} \\quad |Z_n|\\leq t]\\longrightarrow c $$ for $0\\leq c\\leq 1$, we show that the inequality $$ \\ell(2^n-1)\\leq n-1+\\log_2n+C\\sqrt{\\frac{\\log_2n}{\\log \\log_2n}} $$ for some absolute constant $C>0$ with probability that tends to $c$ for $0\\leq c\\leq 1$. This breaks the $O(\\frac{\\log n}{\\log \\log n})$ barrier that is guaranteed to be achieved using the Brauer method in this randomized setting. This result can be seen as injecting probabilistic methods into the theory of addition chains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Theophilus Agama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6njx12ywwgkt7.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "6njx12ywwgkt7",
  "issns": [],
  "landing_url": "https://hal.science/hal-05784795v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
  "licence_label": "CC BY-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05784795/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6njx12ywwgkt7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05784795",
      "source": "hal",
      "url": "https://hal.science/hal-05784795v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "ON A PROBABILISTIC ITERATED FACTOR METHOD",
  "title_text": "ON A PROBABILISTIC ITERATED FACTOR METHOD",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:50Z",
  "url": "https://research.draconitic.com/paper/6njx12ywwgkt7",
  "venue": "HAL",
  "versions": []
}