Discriminators of quadratic polynomials

Abstract

Given $f \in \mathbb{Z}[x]$ and $n \in \mathbb{Z^{+}}$, the $\emph{discriminator}$ $D_f(n)$ is the smallest positive integer $m$ such that $f(1), \ldots, f(n)$ are distinct mod $m$. In a recent paper, Z.-W. Sun proved that $D_f(n) = d^{\lceil \log_d n \rceil}$ if $f(x) = x(dx - 1)$ for $d \in \{2, 3\}$. We extend this result to $d = 2^r$ for any $r \in \mathbb{Z}^{+}$ and find that $D_f(n) = 2^{\lceil \log_2 n \rceil}$ in this case. We also provide more general statements for $d = p^r$, where $p$ is a prime. In addition, we present a potential method for generating prime numbers with discriminators of polynomials which do not always take prime values. Finally, we describe some general statements and possible topics for study about the discriminator of an arbitrary polynomial with integer coefficients.