On a conjecture of Sun involving powers of three

TL;DR

This paper confirms Sun's conjecture that for each positive integer n, the smallest modulus m making the sums a^3 + a distinct modulo m^2 aligns with the least power of 3 not less than the square root of n.

Contribution

The paper proves Sun's conjecture, establishing the exact value of D(n) as the least power of 3 greater than or equal to √n for all n ≥ 2.

Findings

D(n) equals the least power of 3 no less than √n

Confirmed Sun's conjecture for all n ≥ 2

Provides a complete characterization of the minimal modulus

Abstract

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.