{
  "abstract": "A conjecture of Erdős asserts that every odd integer $n > 1$ can be written as $n = 2^a + m$, where $a \\ge 0$ is an integer and $m$ is a squarefree positive integer. We study this conjecture for primes $p \\equiv 7 \\pmod{36}$. Writing $x = (p-1)/6$, the candidate remainders take the shapes $p - 1 = 6x$, $p-2 = 6x-1$, $p-4 = 3(2x-1)$, and $p-8 = 6x-7$, in which the primes $2$ and $3$ are completely controlled: every obstruction to squarefreeness comes from a prime $q \\ge 5$. For such $q$ a separation principle applies --- $q^2$ divides at most one of the remainders, since two of them differ by at most $7 < 25$ --- and it persists across any window of exponents of width less than $20$. Separation collapses the square-divisor problem for the quartic $x(6x-1)(2x-1)(6x-7)$ into independent linear problems, and we deduce unconditionally that the primes of the class representable with $a \\le 3$ have relative density $1 - ρ_4$, where $ρ_4 = 1.8637 \\times 10^{-5}$ is an explicit Euler-type constant; in particular, infinitely many primes of the class are not so representable. Computationally, we verify the conjecture for all primes of the class below $10^8$, always with $a \\le 5$, and we exhibit the eleven primes below $10^8$ requiring $a = 5$. An appendix corrects the error bound in a lemma of Hercher, whose conclusions are unaffected.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07214",
  "author_total": 2,
  "authors": [
    {
      "name": "Shen Harman"
    },
    {
      "name": "Dana Paquin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0f2y38krn24k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07214",
  "fulltext_url": null,
  "id": "f0f2y38krn24k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07214",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221021394",
  "pdf_url": "https://arxiv.org/pdf/2610.07214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0f2y38krn24k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The One-Power Erdős Conjecture for Primes $p \\equiv 7 \\pmod{36}$",
  "title_text": "The One-Power Erdős Conjecture for Primes p ≡ 7 pmod36",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/f0f2y38krn24k",
  "venue": "arXiv",
  "versions": []
}