{
  "id": "1708.00620",
  "version": "v1",
  "published": "2017-08-02T07:20:59.000Z",
  "updated": "2017-08-02T07:20:59.000Z",
  "title": "Differences of Harmonic Numbers and the $abc$-Conjecture",
  "authors": [
    "Natalia da Silva",
    "Serban Raianu",
    "Hector Salgado"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate which numbers other than one can or cannot be written as a difference of harmonic numbers and we look at their connection to the $abc$-conjecture. We find that there are only eleven numbers less than 100 that cannot be written as a difference of harmonic numbers (we call these $ndh$-numbers). The smallest $ndh$-number is 41, which is also Euler's largest lucky number and is a very interesting number. We then show there are infinitely many $ndh$-numbers, some of which are the primes congruent to $41$ modulo $48$. For each Fermat or Mersenne prime we either prove that it is an $ndh$-number or find all ways it can be written as a difference of harmonic numbers. Finally, as suggested by Lenstra in his talk, we interpret Gersonides' theorem as \"The $abc$-conjecture is true on the set of harmonic numbers\" and we expand the set on which the $abc$-conjecture is true by adding to the set of harmonic numbers the following sets (one at a time): a finite set of $ndh$-numbers, the infinite set of primes of the form $48k+41$, the set of Fermat primes, and the set of Mersenne primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-02T07:20:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11A41",
      "11D45"
    ],
    "keywords": [
      "harmonic numbers",
      "difference",
      "conjecture",
      "mersenne prime",
      "eulers largest lucky number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}