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

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 $n$.
Comment: Correct a print mistake in Theorem 1.1