{
  "id": "2212.14139",
  "version": "v1",
  "published": "2022-12-29T00:49:33.000Z",
  "updated": "2022-12-29T00:49:33.000Z",
  "title": "The matrix equation $aX^m+bY^n=cI$ over $M_2(\\mathbb{Z})$",
  "authors": [
    "Hongjian Li",
    "Pingzhi Yuan"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{N}$ be the set of all positive integers and let $a,\\, b,\\, c$ be nonzero integers such that $\\gcd\\left(a,\\, b,\\, c\\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $aX^m+bY^n=cI,\\,X,\\,Y\\in M_2(\\mathbb{Z}),\\, m,\\, n\\in\\mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\\neq YX$ and the solvability of the equation $ax^m+by^n=c,\\, m,\\, n\\in\\mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,\\,X,\\,Y\\in M_2(\\mathbb{Z}),\\,n\\in\\mathbb{N},\\,n\\geq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\\, n\\in\\mathbb{N},\\,n\\geq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,\\,X,\\,Y\\in M_2(\\mathbb{Z})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-29T00:49:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A20",
      "15A24",
      "15B36",
      "11D09",
      "11D41"
    ],
    "keywords": [
      "matrix equation",
      "solvability",
      "quadratic fields",
      "corresponding diophantine equation",
      "nonzero integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}