{
  "id": "1211.4570",
  "version": "v1",
  "published": "2012-11-17T13:22:42.000Z",
  "updated": "2012-11-17T13:22:42.000Z",
  "title": "A congruence modulo $n^3$ involving two consecutive sums of powers and its applications",
  "authors": [
    "Romeo Meštrović"
  ],
  "comment": "16 pages; the manuscript contains 7 new Giuga-Agoh's-like conjectures",
  "categories": [
    "math.NT"
  ],
  "abstract": "For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\\ge 2 $$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\\equiv \\{{array}{ll} 0\\,(\\bmod{\\,n^3}) & {\\rm if}\\,\\,k\\,\\,{\\rm is\\,\\,even\\,\\,or}\\,\\, n\\,\\, {\\rm is\\,\\, odd} & {\\rm or} \\,\\, n\\equiv 0\\,(\\bmod{\\,4}) \\frac{n^3}{2}\\,(\\bmod{\\,n^3}) & {\\rm if}\\,\\,k\\,\\,{\\rm is\\,\\, odd} &,\\,{\\rm and}\\,\\, n\\equiv 2\\,(\\bmod{\\,4}). {array}.$$ The above congruence allows us to state an equivalent formulation of Giuga's conjecture. Moreover, we prove that the first above congruence is satisfied modulo $n^4$ whenever $n\\ge 5$ is a prime number such that $n-1\\nmid 2k-2$. In particular, this congruence arises a conjecture for a prime to be Wolstenholme prime. We also propose several Giuga-Agoh's-like conjectures. Further, we establish two congruences modulo $n^3$ for two binomial type sums involving sums of powers $S_{2i}(n)$ with $i=0,1,...,k$. Furthermore, using the above congruence reduced modulo $n^2$, we obtain an extension of Carlitz-von Staudt result for odd power sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-17T13:22:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11A07",
      "11A51",
      "11B50",
      "11B65",
      "11B68"
    ],
    "keywords": [
      "congruence modulo",
      "consecutive sums",
      "applications",
      "positive integers",
      "odd power sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4570M"
    }
  }
}