{
  "abstract": "For a given number $N$, we consider the problem of computing two integers $1\\leq r,f < N$ such that the set $$\\mathcal{X}(N,r,f) = \\{(a+b)-(f+\\frac{Nr+1}{f}): ab=Nr\\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying $l(Nr) < f \\leq l(Nr+1)$, where $l(x)$ is the largest divisor of $x$ bounded by $\\sqrt{x}$. This requires factoring both $Nr$ and $Nr+1$. We present a simple randomized algorithm that - avoiding factoring - computes pairs $(r,f)$. We give an exact formula for the total number of possible pairs $(r,f)$, and with the aid of empirical data we estimate that the ratio $$\\frac{ϕ(N)-2}{|\\mathfrak{F}(N)|}$$ is approximately about $c*\\log \\log N$. Here, $\\mathfrak{F}(N)$ is the set of unique $r$ appearing among all possible pairs $(r,f)$, $ϕ(.)$ is the Euler's Totient function, and $c$ is a constant equal to 2 for prime $N$ and oscillates much for composite $N$. As a separate and independent case, we study the same problem of computing $(r,f)$ with $r>N$. We present a procedure to find such an $r$, which requires finding the least prime in an arithmetic progression.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08596",
  "author_total": 1,
  "authors": [
    {
      "name": "Srikanth Cherukupally"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmq0y3vmsdq5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08596",
  "fulltext_url": null,
  "id": "cmq0y3vmsdq5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08596",
  "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": "W7212029079",
  "pdf_url": "https://arxiv.org/pdf/2609.08596",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmq0y3vmsdq5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.08596",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08596"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A problem on the largest divisor $d$ of $N$ with $d\\leq \\sqrt{N}$",
  "title_text": "A problem on the largest divisor d of N with d≤ √N",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/cmq0y3vmsdq5d",
  "venue": "arXiv",
  "versions": []
}