{
  "id": "1001.4133",
  "version": "v4",
  "published": "2010-01-23T07:00:54.000Z",
  "updated": "2011-02-02T05:26:29.000Z",
  "title": "Lower bounds on the lengths of double-base representations",
  "authors": [
    "Vassil S. Dimitrov",
    "Everett W. Howe"
  ],
  "comment": "8 pages, LaTeX. Added DOIs for most references; corrected a minor error in arithmetic; made small copy-editing changes. To appear in Proc. Amer. Math. Soc",
  "journal": "Proc. Amer. Math. Soc. 139 (2011), 3423-3430",
  "doi": "10.1090/S0002-9939-2011-10764-0",
  "categories": [
    "math.NT"
  ],
  "abstract": "A double-base representation of an integer n is an expression n = n_1 + ... + n_r, where the n_i are (positive or negative) integers that are divisible by no primes other than 2 or 3; the length of the representation is the number r of terms. It is known that there is a constant a > 0 such that every integer n has a double-base representation of length at most a log n / log log n. We show that there is a constant c > 0 such that there are infinitely many integers n whose shortest double-base representations have length greater than c log n / (log log n log log log n). Our methods allow us to find the smallest positive integers with no double-base representations of several lengths. In particular, we show that 103 is the smallest positive integer with no double-base representation of length 2, that 4985 is the smallest positive integer with no double-base representation of length 3, that 641687 is the smallest positive integer with no double-base representation of length 4, and that 326552783 is the smallest positive integer with no double-base representation of length 5.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-02-02T05:26:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A67",
      "11A63"
    ],
    "keywords": [
      "smallest positive integer",
      "lower bounds",
      "log log log",
      "shortest double-base representations",
      "length greater"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}