{
  "id": "2005.05214",
  "version": "v1",
  "published": "2020-05-11T16:03:21.000Z",
  "updated": "2020-05-11T16:03:21.000Z",
  "title": "On a class of Labesgue-Ramanujan-Nagell equations",
  "authors": [
    "Azizul Hoque"
  ],
  "comment": "12 Pages. Comments are most welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "We investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\\geq 1, m\\geq 0$ and $n\\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\\not\\equiv 3 \\pmod 4$. We first solve this equation for prime $n$, under the condition $n\\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field $\\mathbb{Q}(\\sqrt{-cd})$. We then completely solve this equation for both $c$ and $d$ primes, under the assumption $\\gcd(n, h(-cd))=1$. We also completely solve this equation for $c=1$ and $d\\equiv1 \\pmod 4$, under the condition $\\gcd(n, h(-d))=1$. For some fixed values of $c$ and $d$, we derive some results concerning the solvability of this equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-11T16:03:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41",
      "11B39",
      "11Y50"
    ],
    "keywords": [
      "labesgue-ramanujan-nagell equations",
      "diophantine equation",
      "class number",
      "coprime positive integers",
      "quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}