Arithmetic properties of polynomials

Abstract

First, we prove that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$ 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 $$\begin{aligned} {\left\{ \begin{array}{ll} (n^2-m^2)(4mnk(k+a+1)+a(m^2+2mn-n^2)) &{}\equiv 0\pmod (m^2+2mn-n^2)((m^2-2mn-n^2)k(k+a+1)-2amn) &{}\equiv 0\pmod {(m^2+n^2)^2}, \end{array}\right. } \end{aligned}$$ where $$k\equiv 0\pmod {4}$$ when a is even, and $$k\equiv 2\pmod {4}$$ when a is odd. Third, we get that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=\frac{f(r)}{f(s)} \end{aligned}$$ 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 $$a\ne b$$ . Finally, we raise some related questions.