  Given a positive integer $n\ge 2$, let $D(n)$ denote the smallest positive
integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m^2$.
A conjecture of Z.-W. Sun states that $D(n)=3^k$, where $3^k$ is the least
power of $3$ no less than $\sqrt{n}$. The purpose of this paper is to confirm
this conjecture.
