{
  "id": "2301.09263",
  "version": "v1",
  "published": "2023-01-23T04:25:08.000Z",
  "updated": "2023-01-23T04:25:08.000Z",
  "title": "On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields",
  "authors": [
    "Narasimha Kumar",
    "Satyabrat Sahoo"
  ],
  "comment": "Submitted for publication; Any comments are welcome. arXiv admin note: text overlap with arXiv:2207.10930",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \\in \\mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \\in \\mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-23T04:25:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11R80",
      "11F80",
      "11G05",
      "11R04"
    ],
    "keywords": [
      "totally real fields",
      "odd integer",
      "prime exponent",
      "diophantine equation",
      "non-trivial primitive solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}