{
  "id": "2006.11634",
  "version": "v1",
  "published": "2020-06-20T18:54:20.000Z",
  "updated": "2020-06-20T18:54:20.000Z",
  "title": "A Fractional $3n+1$ Conjecture",
  "authors": [
    "Éric Brier",
    "Rémi Géraud-Stewart",
    "David Naccache"
  ],
  "categories": [
    "math.DS",
    "cs.FL"
  ],
  "abstract": "In this paper we introduce and discuss the sequence of \\emph{real numbers} defined as $u_0 \\in \\mathbb R$ and $u_{n+1} = \\Delta(u_n)$ where \\begin{equation*} \\Delta(x) = \\begin{cases} \\frac{x}{2} &\\text{if } \\operatorname{frac}(x)<\\frac{1}{2} \\\\[4px] \\frac{3x+1}{2} & \\text{if } \\operatorname{frac}(x)\\geq\\frac{1}{2} \\end{cases} \\end{equation*} This sequence is reminiscent of the famous Collatz sequence, and seems to exhibit an interesting behaviour. Indeed, we conjecture that iterating $\\Delta$ will eventually either converge to zero, or loop over sequences of real numbers with integer parts $1,2,4,7,11,18,9,4,7,3,5,9,4,7,11,18,9,4,7,3,6,3,1,2,4,7,3,6,3$. We prove this conjecture for $u_0 \\in [0, 100]$. Extending the proof to larger fixed values seems to be a matter of computing power. The authors pledge to offer a reward to the first person who proves or refutes the conjecture completely -- with a proof published in a serious refereed mathematical conference or journal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-20T18:54:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conjecture",
      "fractional",
      "authors pledge",
      "larger fixed values",
      "famous collatz sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}