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<p>Given a positive integer <i>n</i> ≥ 2, let <i>D</i>(<i>n</i>) denote the smallest positive integer <i>m</i> such that <i>a</i><sup>3</sup> + <i>a</i> (<i>a</i> = 1, ⋯, <i>n</i>) are pairwise distinct modulo <i>m</i><sup>2</sup>. A conjecture of Sun states that <i>D</i>(<i>n</i>) = 3<sup><i>k</i></sup>, where 3<sup><i>k</i></sup> is the least power of 3 no less than <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\sqrt n}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msqrt> <mi>n</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation>. The purpose of this paper is to confirm this conjecture.</p>

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On a Conjecture of Sun Involving Powers of Three

  • 期刊论文

  • 发表日期：

    2026年7月13日

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  • Quan-Hui Yang,
  • Lilu Zhao

摘要

Given a positive integer n ≥ 2, let D(n) denote the smallest positive integer m such that a3 + a (a = 1, ⋯, n) are pairwise distinct modulo m2. A conjecture of Sun states that D(n) = 3k, where 3k is the least power of 3 no less than \({\sqrt n}\) n . The purpose of this paper is to confirm this conjecture.