连续二次剩余的分布性质及其应用

Xiao Wang; Ai Hua Li

doi:10.12386/A20210160

连续二次剩余的分布性质及其应用

Translated title of the contribution: Distribution Properties and Applications of Consecutive Quadratic Residues

Xiao Wang, Ai Hua Li

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Consider any prime number p. In this paper, we use analytic methods, properties of Legendre’s symbol modulo p, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo p. We provide exact formulas for the numbers S₁(p) and S₂(p) of 3-CQRs and 3-CQNs when p ≡ 3 or 7 (mod 8). Asymptotic formulas are given for p ≡ 1 or 5 (mod 8). Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given.

Translated title of the contribution	Distribution Properties and Applications of Consecutive Quadratic Residues
Original language	Chinese (Simplified)
Pages (from-to)	389-398
Number of pages	10
Journal	Acta Mathematica Sinica, Chinese Series
Volume	66
Issue number	2
DOIs	https://doi.org/10.12386/A20210160
State	Published - 15 Mar 2023

Keywords

Legendre’s symbol
character sums
quadratic non-residues
quadratic residues

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abstract = "Consider any prime number p. In this paper, we use analytic methods, properties of Legendre{\textquoteright}s symbol modulo p, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo p. We provide exact formulas for the numbers S1(p) and S2(p) of 3-CQRs and 3-CQNs when p ≡ 3 or 7 (mod 8). Asymptotic formulas are given for p ≡ 1 or 5 (mod 8). Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given.",

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T1 - 连续二次剩余的分布性质及其应用

AU - Wang, Xiao

AU - Li, Ai Hua

PY - 2023/3/15

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N2 - Consider any prime number p. In this paper, we use analytic methods, properties of Legendre’s symbol modulo p, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo p. We provide exact formulas for the numbers S1(p) and S2(p) of 3-CQRs and 3-CQNs when p ≡ 3 or 7 (mod 8). Asymptotic formulas are given for p ≡ 1 or 5 (mod 8). Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given.

AB - Consider any prime number p. In this paper, we use analytic methods, properties of Legendre’s symbol modulo p, and estimation for character sums to study distribution properties of triples of consecutive quadratic residues (named 3-CQR) and consecutive quadratic non-residues (3-CQN) modulo p. We provide exact formulas for the numbers S1(p) and S2(p) of 3-CQRs and 3-CQNs when p ≡ 3 or 7 (mod 8). Asymptotic formulas are given for p ≡ 1 or 5 (mod 8). Similarly, triples of quadratic residues with equal distance 2 are investigated and corresponding enumeration formulas are given.

KW - Legendre’s symbol

KW - character sums

KW - quadratic non-residues

KW - quadratic residues

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JO - Acta Mathematica Sinica, Chinese Series

JF - Acta Mathematica Sinica, Chinese Series

IS - 2

ER -

连续二次剩余的分布性质及其应用

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