{
  "id": "1706.03433",
  "version": "v1",
  "published": "2017-06-12T01:38:36.000Z",
  "updated": "2017-06-12T01:38:36.000Z",
  "title": "Arithmetic properties of polynomials",
  "authors": [
    "Yong Zhang",
    "Zhongyan Shen"
  ],
  "comment": "18 pages. Submitted for publication",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, first, we prove that the Diophantine system \\[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\\equiv 0,1,4\\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \\[\\begin{cases} \\begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\\equiv0\\pmod{(m^2+n^2)^2},\\\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\\equiv0 \\pmod{(m^2+n^2)^2}, \\end{split} \\end{cases}\\] where $k\\equiv0\\pmod{4}$ when $a$ is even, and $k\\equiv2\\pmod{4}$ when $a$ is odd. Third, we get that the Diophantine system \\[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=\\frac{f(r)}{f(s)}\\] has a five-parameter rational solution for $f(X)=X(X+a)$ with nonzero rational number $a$ and infinitely many nontrivial rational parametric solutions for $f(X)=X(X+a)(X+b)$ with nonzero integers $a,b$ and $a\\neq b$. At last, we raise some related questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-12T01:38:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D25",
      "11D72",
      "11G05"
    ],
    "keywords": [
      "arithmetic properties",
      "diophantine system",
      "nonzero integers",
      "polynomials",
      "nontrivial rational parametric solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}