On two conjectures related to cubic residues

Xiaopeng Zhao; Zhenfu Cao

doi:10.1007/s13226-024-00626-z

On two conjectures related to cubic residues

Xiaopeng Zhao^*
, Zhenfu Cao

^*此作品的通讯作者

软件工程学院

Donghua University

科研成果: 期刊稿件 › 文章 › 同行评审

摘要

In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding S₃(p) which is the number of all integers a∈{1,2,…,p-1} such that a+a^-1 and a-a^-1 are both cubic residues modulo a prime p≡1(mod3). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for S₃(p) when 2 is a cubic non-residue modulo p.

源语言	英语
期刊	Indian Journal of Pure and Applied Mathematics
DOI	https://doi.org/10.1007/s13226-024-00626-z
出版状态	已接受/待刊 - 2024

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title = "On two conjectures related to cubic residues",

abstract = "In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding S3(p) which is the number of all integers a∈\{1,2,…,p-1\} such that a+a-1 and a-a-1 are both cubic residues modulo a prime p≡1(mod3). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for S3(p) when 2 is a cubic non-residue modulo p.",

keywords = "11A07, 11A15, Calculation formula, Cubic residue character, Cubic residuosity, Gauss sums",

author = "Xiaopeng Zhao and Zhenfu Cao",

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year = "2024",

doi = "10.1007/s13226-024-00626-z",

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AU - Cao, Zhenfu

N1 - Publisher Copyright: © The Indian National Science Academy 2024.

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N2 - In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding S3(p) which is the number of all integers a∈{1,2,…,p-1} such that a+a-1 and a-a-1 are both cubic residues modulo a prime p≡1(mod3). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for S3(p) when 2 is a cubic non-residue modulo p.

AB - In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding S3(p) which is the number of all integers a∈{1,2,…,p-1} such that a+a-1 and a-a-1 are both cubic residues modulo a prime p≡1(mod3). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for S3(p) when 2 is a cubic non-residue modulo p.

KW - 11A07

KW - 11A15

KW - Calculation formula

KW - Cubic residue character

KW - Cubic residuosity

KW - Gauss sums

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DO - 10.1007/s13226-024-00626-z

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AN - SCOPUS:85197481818

SN - 0019-5588

JO - Indian Journal of Pure and Applied Mathematics

JF - Indian Journal of Pure and Applied Mathematics

ER -

On two conjectures related to cubic residues

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