{
  "abstract": "Paul Erdős conjectured that the only powers of two whose ternary expansions contain no digit 2 are 1, 4, and 256. Despite extensive computational verification, the conjecture remains open. We reformulate the problem using generalised base-4 repunits. Any admissible exponent is even, say \\(n=2s\\), and \\[ 2^{2s}=4^s=3R_s(4)+1,\\qquad R_s(4)=\\frac{4^s-1}{3}. \\] Thus \\(2^{2s}\\) omits the ternary digit 2 if and only if \\(R_s(4)\\) does. For each depth \\(m\\), the map \\(Φ_m:s\\bmod 3^m\\mapsto R_s(4)\\bmod 3^m\\) is a permutation of the residue classes modulo \\(3^m\\). Hence every prescribed \\(m\\)-digit ternary suffix has a unique canonical exponent seed modulo \\(3^m\\), and all exponents producing that suffix form an arithmetic progression. The lifting identity \\[ R_{r+q3^m}(4)\\equiv R_r(4)+q3^m\\pmod{3^{m+1}} \\] gives three lifts at each depth, exactly two of which preserve admissibility. The finite maps assemble into an isometric homeomorphism of the 3-adic integers conjugating \\(x \\mapsto x+1\\) with \\(y \\mapsto 4y+1\\). The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03789",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael A. Idowu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5pmpt5ew2bm5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03789",
  "fulltext_url": null,
  "id": "a5pmpt5ew2bm5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03789",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220690427",
  "pdf_url": "https://arxiv.org/pdf/2610.03789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5pmpt5ew2bm5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Repunit Coordinates for the Erdős Ternary-Digit Problem",
  "title_text": "Repunit Coordinates for the Erdős Ternary-Digit Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/a5pmpt5ew2bm5",
  "venue": "arXiv",
  "versions": []
}