{
  "id": "2107.03908",
  "version": "v1",
  "published": "2021-07-08T15:39:57.000Z",
  "updated": "2021-07-08T15:39:57.000Z",
  "title": "On some Generalized Fermat Equations of the form $x^2+y^{2n} = z^p$",
  "authors": [
    "Philippe Michaud-Rodgers"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The primary aim of this paper is to study the generalized Fermat equation \\[ x^2+y^{2n} = z^{3p} \\] in coprime integers $x$, $y$, and $z$, where $n \\geq 2$ and $p$ is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a complete resolution of this equation in the case $p=7$, and obtain an asymptotic result for fixed $p$. Additionally, using similar techniques, we solve a second equation, namely $x^{2\\ell}+y^{2m} = z^{17}$, for primes $\\ell,m \\ne 5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-08T15:39:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11F80",
      "11G05",
      "11F41"
    ],
    "keywords": [
      "generalized fermat equation",
      "hilbert cuspidal eigenforms",
      "complete resolution",
      "second equation",
      "similar techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}