{
  "id": "1801.01542",
  "version": "v1",
  "published": "2017-12-20T18:45:43.000Z",
  "updated": "2017-12-20T18:45:43.000Z",
  "title": "Congruences of Power Sums",
  "authors": [
    "Nicholas J. Newsome",
    "Maria S. Nogin",
    "Adnan H. Sabuwala"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\\dots +p^n \\equiv\\begin{cases} -1 \\text{ mod } p, & \\text{ if } \\ p-1 \\ | \\ n; 0 \\text{ mod } p, & \\text{ if } \\ p-1 \\ \\not| \\ n, \\end{cases}$ where $n\\in{\\mathbb N}$ and $p$ is prime. We extend this congruence, in particular, to the case when $p$ is any power of a prime. We also show that the sequence $(S_n(m) \\text{ mod } k )_{m \\geq 1}$ is periodic and determine its period.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-20T18:45:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B50",
      "11A25",
      "11B83"
    ],
    "keywords": [
      "power sums",
      "congruence",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}