Discriminators of quadratic polynomials

TL;DR

This paper studies the discriminator of quadratic polynomials, extending previous results to powers of two and prime powers, and explores methods for generating primes using polynomial discriminators.

Contribution

It generalizes the known discriminator results for specific quadratic polynomials to broader classes involving prime powers and discusses potential prime generation methods.

Findings

Discriminator for $f(x) = x(d x - 1)$ with $d=2^r$ is $2^{ ceil \, ext{log}_2 n ceil}$.

Extended the discriminator results to prime power cases $d=p^r$.

Proposed a method for generating primes via polynomial discriminators.

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.