{
  "id": "0712.2365",
  "version": "v2",
  "published": "2007-12-14T15:05:49.000Z",
  "updated": "2008-04-14T14:49:54.000Z",
  "title": "Ternary cyclotomic polynomials having a large coefficient",
  "authors": [
    "Yves Gallot",
    "Pieter Moree"
  ],
  "comment": "19 pages, 6 tables, to appear in Crelle's Journal. Revised version with many small changes",
  "journal": "J. Reine Angew. Math. 632 (2009), 105-125",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\\Phi_n(x)$, satisfies $|a_n(k)|\\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\\ge 11$. We also prove that given any $\\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\\epsilon)p_j$ for $j\\ge 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-14T14:49:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "11C08"
    ],
    "keywords": [
      "ternary cyclotomic polynomials",
      "large coefficient",
      "th cyclotomic polynomial",
      "sister marion beiter",
      "beiters conjecture"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.2365G"
    }
  }
}