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<p>The main purpose of this paper is using analytic methods, an estimate for character sums, and the properties of the third-order characters to study the calculating problems of a certain cubic residues modulo <i>p</i>, an odd prime, and to give an exact computational formula for its counting function. This solves two conjectures proposed by X. Yuan and W. Zhang in [On cubic residues and related problems, <i>Indian J. Pure Appl. Math.</i>, 54(3):806–815, 2023].</p>

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Two conjectures related to cubic residues

  • 期刊论文

  • 发表日期：

    2026年2月6日

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  • Wenpeng Zhang,
  • Xiaoling Xu

摘要

The main purpose of this paper is using analytic methods, an estimate for character sums, and the properties of the third-order characters to study the calculating problems of a certain cubic residues modulo p, an odd prime, and to give an exact computational formula for its counting function. This solves two conjectures proposed by X. Yuan and W. Zhang in [On cubic residues and related problems, Indian J. Pure Appl. Math., 54(3):806–815, 2023].