{
  "id": "2102.07977",
  "version": "v1",
  "published": "2021-02-16T07:00:22.000Z",
  "updated": "2021-02-16T07:00:22.000Z",
  "title": "On the Diophantine equation $cx^2+p^{2m}=4y^n$",
  "authors": [
    "Kalyan Chakraborty",
    "Azizul Hoque",
    "Kotyada Srinivas"
  ],
  "comment": "12 pages. To appear in `Results in Mathematics'",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\\mathbb{Q}(\\sqrt{-c})$. In this paper, we consider the Diophantine equation $$cx^2+p^{2m}=4y^n,~~x,y\\geq 1, m\\geq 0, n\\geq 3, \\gcd(x,y)=1, \\gcd(n,2h(-c))=1,$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-16T07:00:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41",
      "11Y50"
    ],
    "keywords": [
      "diophantine equation",
      "imaginary quadratic field",
      "class number",
      "square-free positive integer",
      "integer solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}