{
  "id": "2201.00127",
  "version": "v2",
  "published": "2022-01-01T06:17:29.000Z",
  "updated": "2022-12-12T14:01:37.000Z",
  "title": "Extremal sequences related to the Jacobi symbol",
  "authors": [
    "Santanu Mondal",
    "Krishnendu Paul",
    "Shameek Paul"
  ],
  "comment": "15 pages. arXiv admin note: substantial text overlap with arXiv:2111.14477",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For a weight-set $A\\subseteq \\mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of consecutive terms. A sequence of length $C_A(n)-1$ in $\\mathbb Z_n$ which does not have any $A$-weighted zero-sum subsequence of consecutive terms will be called a $C$-extremal sequence for $A$. Let $\\big(\\frac{x}{n}\\big)$ denote the Jacobi symbol of $x\\in\\mathbb Z_n$. We characterize the $C$-extremal sequences for the weight-set $S(n)=\\big\\{\\,x\\in U(n):\\big(\\frac{x}{n}\\big)=1\\,\\big\\}$ and for the weight-set $L(n;p)=\\big\\{\\,x\\in U(n):\\big(\\frac{x}{n}\\big)=\\big(\\frac{x}{p}\\big)\\,\\big\\}$ where $p$ is a prime divisor of $n$. We can define $D$-extremal sequences for these weight-sets in a way analogous to the definition of $C$-extremal sequences. We also characterize these sequences.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-12T14:01:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50"
    ],
    "keywords": [
      "extremal sequence",
      "jacobi symbol",
      "weighted zero-sum subsequence",
      "weight-set",
      "smallest natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}