{
  "id": "2407.03822",
  "version": "v1",
  "published": "2024-07-04T10:46:15.000Z",
  "updated": "2024-07-04T10:46:15.000Z",
  "title": "Some Diophantine equations involving arithmetic functions and Bhargava factorials",
  "authors": [
    "Daniel M. Baczkowski",
    "Saša Novaković"
  ],
  "comment": "8 pages, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "F. Luca proved for any fixed rational number $\\alpha>0$ that the Diophantine equations of the form $\\alpha\\,m!=f(n!)$, where $f$ is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions $(m,n)$. In this paper we generalize the mentioned result and show that Diophantine equations of the form $\\alpha\\,m_1!\\cdots m_r!=f(n!)$ have finitely many integer solutions, too. In addition, we do so by including the case $f$ is the sum of $k$\\textsuperscript{th} powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-04T10:46:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11A41",
      "11D99"
    ],
    "keywords": [
      "diophantine equations",
      "bhargava factorials",
      "arithmetic functions",
      "integer solutions",
      "divisor sum function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}