{
  "id": "math/0103054",
  "version": "v1",
  "published": "2001-03-08T00:15:08.000Z",
  "updated": "2001-03-08T00:15:08.000Z",
  "title": "Lower bounds for the total stopping time of 3X+1 iterates",
  "authors": [
    "David Applegate",
    "Jeffrey C. Lagarias"
  ],
  "comment": "21 pages latex, 1 figure",
  "journal": "Math. Comp. 72 (2003), 1035--1049.",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time \\sigma_\\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \\infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with \\sigma_\\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have \\sigma_\\infty (n) > 6.95212 \\log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-03-08T00:15:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "11Y16"
    ],
    "keywords": [
      "lower bounds",
      "positive integer",
      "finite total stopping time",
      "large computation",
      "larger computation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Mathematics of Computation",
      "year": 2003,
      "volume": 72,
      "pages": 1035
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003MaCom..72.1035A"
    }
  }
}