{
  "abstract": "We prove that each of the inequalities $P^{+}(p+1)>P^{+}(p-1)$ and $P^{+}(p-1)>P^{+}(p+1)$ holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35610",
  "author_total": 1,
  "authors": [
    {
      "name": "Genheng Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atr09k0brcdvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35610",
  "fulltext_url": null,
  "id": "atr09k0brcdvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35610",
  "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": "W7214784663",
  "pdf_url": "https://arxiv.org/pdf/2609.35610",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atr09k0brcdvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.35610",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35610"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the largest prime factors of $p-1$ and $p+1$",
  "title_text": "On the largest prime factors of p−1 and p+1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/atr09k0brcdvy",
  "venue": "arXiv",
  "versions": []
}