{
  "abstract": "For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for $x\\rightarrow\\infty$, one has $\\#\\{p\\leq x:P^+(p-1)<P^+(p+1)\\}\\sim π(x)/2$. In this paper, we show the following results. For any $3\\leq y\\ll x^{o(1)}$, for $x\\rightarrow\\infty$, we have \\begin{align*} &\\#\\{n\\leq x:P^+_y(n)<P^+_y(n+1)\\}\\sim \\frac{1}{2}x, \\\\&\\#\\{p\\leq x:P^+_y(p-1)<P^+_y(p+1)\\}\\sim \\frac{1}{2}π(x), \\end{align*} where the first result extends a previous range $3\\leq y\\ll \\exp(\\log x/(100\\log\\log x))$ due to Rivat (2001). For any $0<α<17/32$, let $y=x^α$. Then there exists $h(α)>0$ such that \\begin{align*} \\#\\{p\\leq x:P_y^+(p-1)<P_y^+(p+1)\\}\\geq(h(α)+o(1))π(x). \\end{align*} In particular, the function $h$ satisfies $\\lim_{α\\rightarrow 0^+}h(α)=1/2$. Similar result also holds for $\\#\\{n\\leq x:P_y^+(n)<P_y^+(n+1)\\}$, which improves a previous result $\\lim_{α\\rightarrow 0^+} h(α)=1/4$ due to Wang (2019).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15166",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhiyuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94d1xyh9sxgne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15166",
  "fulltext_url": null,
  "id": "94d1xyh9sxgne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15166",
  "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": "W7213342621",
  "pdf_url": "https://arxiv.org/pdf/2609.15166",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94d1xyh9sxgne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.15166",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15166"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the largest prime factors less than $y$ of consecutive shifted primes",
  "title_text": "On the largest prime factors less than y of consecutive shifted primes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/94d1xyh9sxgne",
  "venue": "arXiv",
  "versions": []
}