{
  "abstract": "For primes $n\\equiv1\\pmod{24}$, we study the depth of explicit, factorization-free searches for a decomposition of $4/n$ into three positive unit fractions. Let $E_2(N;J)$ be the set of such primes $n\\le N$ for which no coprime pair $1\\le u,a\\le J$ admits a divisor $s\\mid an+u$ with $s\\equiv-1\\pmod{4ua}$. For every fixed $J$, we prove $|E_2(N;J)|\\ll_J N/(\\log N)^{1+A(J)/2}$, where $A(J)$ counts the coprime pairs in $[1,J]^2$. The proof uses a fixed-point-free involution and identifies the exact dimension of the covering. We also establish a dual Type I divisor criterion and an unconditional obstruction: at the fixed shift $c=7$, both branches of the cofactor criterion fail for $\\asymp N/(\\log N)^{3/2}$ primes. At growing depth, an exact reciprocal-totient asymptotic with leading constant $9/(2π^2)$ improves the explicit large-sieve bounds to $O(N\\exp(-0.0942(\\log N)^{2/3}))$ at cutoff $p>J^5$, and to $O_\\varepsilon(N\\exp(-(0.1093-\\varepsilon)(\\log N)^{2/3}))$ at cutoff $p>J^{4+\\varepsilon}$ for fixed sufficiently small $\\varepsilon>0$. Finally, combining a truncated Type I pass with a sparse residual search yields a complete, factorization-free batch decision procedure for all primes in $[N,2N]$, using $O(N^2(\\log N)^2\\exp(-c_0(\\log N)^{2/3}))=o(N^2)$ arithmetic operations for some $c_0>0$. The conjecture itself remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24035",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Dahan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bpwtsvtvbf0ac.bib",
  "cited_by": 2,
  "doi": "10.48550/arxiv.2608.24035",
  "fulltext_url": null,
  "id": "bpwtsvtvbf0ac",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24035",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7203781147",
  "pdf_url": "https://arxiv.org/pdf/2608.24035",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bpwtsvtvbf0ac.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2608.24035",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24035"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sieve dimension and search depth for the Erdős-Straus conjecture, $n \\equiv 1 \\pmod{24}$",
  "title_text": "Sieve dimension and search depth for the Erdős-Straus conjecture, n ≡ 1 pmod24",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/bpwtsvtvbf0ac",
  "venue": "arXiv",
  "versions": []
}