{
  "id": "2012.15401",
  "version": "v1",
  "published": "2020-12-31T02:18:10.000Z",
  "updated": "2020-12-31T02:18:10.000Z",
  "title": "On some conjectures of exponential Diophantine equations",
  "authors": [
    "Hairong Bai"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a, b, c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{r}, r\\in Z^{+}, 2\\mid r$ with $b$ even. That is $$a=\\mid Re(m+n\\sqrt{-1})^{r}\\mid, b=\\mid Im(m+n\\sqrt{-1})^{r}\\mid, c=m^{2}+n^{2},$$ where $m, n$ are positive integers with $m>n, m-n\\equiv1(mod 2),$ gcd$(m, n)=1.$ $(x, y, z)= (2, 2, r)$ is called the trivial solution of the equation. In this paper we prove that the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r\\equiv 2(mod 4), m\\equiv 3(mod 4), m>\\max\\{n^{10.4\\times10^{11}\\log(5.2\\times10^{11}\\log n)}, 3e^{r}, 70.2nr\\}.$$ Especially the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r=2, m\\equiv 3(mod 4), m>n^{10.4\\times10^{11}\\log(5.2\\times10^{11}\\log n)}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-31T02:18:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential diophantine equation",
      "conjectures",
      "nontrivial solutions",
      "relatively prime positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}