{
  "abstract": "In this paper, we consider the asymptotic density of $T_c(x):=\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\\rightarrow\\infty$, for any $0<c\\leq 1/2$, one has \\begin{align*} \\mathop{\\lim\\inf}_{x\\rightarrow\\infty}\\frac{T_c(x)}{π(x)}&\\geq\\max\\left(1-\\frac{16}{5}ρ\\left(\\frac{1}{c}\\right),1-c+g(c)\\right), \\end{align*} where $g(c)>0$ for $11/32<c\\leq1/2$, and $g(c)=0$ for $0<c\\leq11/32$. In particular, we have $g(1/2)>0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed \\begin{align*} \\mathop{\\lim\\inf}_{x\\rightarrow\\infty}\\frac{T_c(x)}{π(x)}&\\geq\\max\\left(1-4ρ\\left(\\frac{1}{c}\\right),1-c\\right), \\end{align*} for $0<c\\leq 1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24347",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhiyuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6w0xt1fb647cq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24347",
  "fulltext_url": null,
  "id": "6w0xt1fb647cq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24347",
  "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": "W7214054474",
  "pdf_url": "https://arxiv.org/pdf/2609.24347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6w0xt1fb647cq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The lower bound of shifted primes with large prime factors",
  "title_text": "The lower bound of shifted primes with large prime factors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6w0xt1fb647cq",
  "venue": "arXiv",
  "versions": []
}