{
  "id": "2001.09810",
  "version": "v1",
  "published": "2020-01-23T06:04:18.000Z",
  "updated": "2020-01-23T06:04:18.000Z",
  "title": "Generator of Pythagorean triples and Je$\\acute{s}$manowicz conjecture",
  "authors": [
    "Nainrong Feng"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a,b,c$ be relatively prime positive integers such that $a^2+b^2=c^2, 2|b$. In this paper, we show that Pythagorean triples $(a, b,c)$ must satisfy $abc\\equiv{0\\; (\\mod3\\cdot{4}\\cdot{5}})$ and $c\\neq{0\\; (\\mod{3}})$, and we also prove that for $(a,b,c)\\in\\{(a,b,c)|a\\equiv{0\\;(\\mod{3}}),b\\equiv{0\\;(\\mod{4}}),c\\equiv{0\\; (\\mod{5}})\\}\\bigcup\\{(a,b,c)|b\\equiv{0\\;(\\mod{12}}),c\\equiv{0\\;(\\mod{5}})\\}$, the only solution of $$a^x+b^y=c^z\\qquad{z},y,z\\in{N}$$ in positive integers is $(x, y, z) = (2, 2,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-23T06:04:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61"
    ],
    "keywords": [
      "pythagorean triples",
      "manowicz conjecture",
      "relatively prime positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}