{
  "id": "1312.7295",
  "version": "v2",
  "published": "2013-12-27T16:45:19.000Z",
  "updated": "2015-05-13T14:52:36.000Z",
  "title": "An asymptotic formula for Goldbach's conjecture with monic polynomials in $\\mathbb{Z}[θ][x]$",
  "authors": [
    "Abílio Lemos",
    "Anderson L. A. Araujo"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider $D=\\mathbb{Z}[\\theta]$, where $$\\theta= \\sqrt{-k} \\,\\,\\,\\, \\mbox{if}\\;\\;\\;-k\\not\\equiv 1 \\;(\\mbox{mod}\\;4)\\,\\,\\,\\,\\mbox{or}\\,\\,\\,\\, \\theta=\\frac{\\sqrt{-k}+1}{2} \\,\\,\\,\\, \\mbox{if}\\;\\;\\;-k\\equiv 1 \\;(\\mbox{mod}\\;4),$$ $k\\geq 2$ is a squarefree integer, and we proved that the number $R(y)$ of representations of a monic polynomial $f(x)\\in \\mathbb{Z}[\\theta][x]$, of degree $d\\geq 1$, as a sum of two monic irreducible polynomials $g(x)$ and $h(x)$ in $\\mathbb{Z}[\\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic to $(4y)^{2d-2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-27T16:45:19.000Z",
      "title": "An asymptotic formula for Goldbach's conjecture with monic polynomials in Z[theta][x]",
      "abstract": "We say that $D$ satisfies the property (GC) if \\[ \\begin{array}{l} \\textup{Every element of} \\ D[x] \\ \\textup{of degree} \\ d\\geq 1 \\textup{can be written as the sum of two irreducibles in }\\,D[x]. \\end{array} \\] In 1965 Hayes \\cite{Hayes} showed that $\\mathbb{Z}$ satisfies the property (GC). In 2011 Pollack showed that $D$ satisfies the property (GC), where $D$ is any integral domain. In this note, we considere $D=\\mathbb{Z}[\\theta]$, where $$\\theta=\\left\\{\\begin{array}{ccc} \\sqrt{-k} & \\mbox{if}\\;\\;\\;-k\\not\\equiv 1 \\;(\\mbox{mod}\\;4) \\frac{\\sqrt{-k}+1}{2} & \\mbox{if}\\;\\;\\;-k\\equiv 1 \\;(\\mbox{mod}\\;4) \\end{array}\\right.,$$ $k$ is a squarefree integer and $k\\geq 2$, and we proved that the number $R(y)$ of representation of a monic polynomial $f(x)\\in \\mathbb{Z}[\\theta][x]$ as a sum of two irreducible monic polynomials $g(x)$ and $h(x)$ in $\\mathbb{Z}[\\theta][x]$, with the coefficients of $g(x)$ and $h(x)$ bounded in complex modulus by $y$, is asymptotic to $(4y)^{2d-2}$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-05-13T14:52:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "goldbachs conjecture",
      "asymptotic formula",
      "squarefree integer",
      "integral domain",
      "irreducible monic polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7295L"
    }
  }
}