{
  "abstract": "An addition chain of length $h$ that leads to an integer $n\\geq 2$ is a strictly increasing sequence of positive integers $s_0=1,s_1=2,\\ldots,s_h=n$ such that $s_i=s_j+s_k$ with $i>j\\geq k\\geq 0$ for each $i\\in \\{1,\\ldots,h\\}$. We denote the minimal length of an addition chain that leads to an integer target $\\cdot$ by $\\ell(\\cdot)$. In this paper, we introduce the concept of \\emph{quasi closed} addition chains and show that numbers $n\\geq 2$ that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality $$ \\ell(2^n-1)\\leq n-1+\\ell(n). $$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Theophilus Agama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egee8wcwrsqvv.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "egee8wcwrsqvv",
  "issns": [],
  "landing_url": "https://hal.science/hal-05755806v1",
  "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-05755806/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egee8wcwrsqvv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:36Z",
      "native_id": "hal-05755806",
      "source": "hal",
      "url": "https://hal.science/hal-05755806v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "ON QUASI CLOSED CHAINS AND QUASI COMPLETE NUMBERS",
  "title_text": "ON QUASI CLOSED CHAINS AND QUASI COMPLETE NUMBERS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:36Z",
  "url": "https://research.draconitic.com/paper/egee8wcwrsqvv",
  "venue": "HAL",
  "versions": []
}