On a discriminator for the polynomial $f(x)=x^3+x$

TL;DR

This paper investigates the minimal modulus that makes the values of the polynomial $x^3 + x$ distinct modulo that number for all integers from 1 to n, providing a comprehensive determination of this discriminator function.

Contribution

It explicitly determines the value of $ riangle(n)$ for all positive integers n, solving a specific problem in polynomial discriminators.

Findings

Explicit formulas for $ riangle(n)$ for all n

Identification of the minimal modulus for polynomial value distinctness

Complete characterization of the discriminator for $f(x)=x^3+x$

Abstract

Let $\Delta(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m$. The purpose of this paper is to determine $\Delta(n)$ for all positive integers $n$.