A new theorem on quadratic residues modulo primes

Qing-Hu Hou; Hao Pan; Zhi-Wei Sun

doi:10.5802/crmath.371

Number theory

A new theorem on quadratic residues modulo primes

Qing-Hu Hou ¹; Hao Pan ²; Zhi-Wei Sun ³

¹ School of Mathematics, Tianjin University, Tianjin 300350, People’s Republic of China
² School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, People’s Republic of China
³ Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1065-1069

Abstract

Let $p > 3$ be a prime, and let $(\frac{\cdot}{p})$ be the Legendre symbol. Let $b \in ℤ$ and $ε \in {\pm 1}$ . We mainly prove that

|\{N_{p} (a, b) : 1 < a < p and (\frac{a}{p}) = ε\}| = \frac{3 - (\frac{- 1}{p})}{2},

where $N_{p} (a, b)$ is the number of positive integers $x < p / 2$ with ${x^{2} + b}_{p} > {a x^{2} + b}_{p}$ , and ${m}_{p}$ with $m \in ℤ$ is the least nonnegative residue of $m$ modulo $p$ .

Received: 2022-03-08
Accepted: 2022-04-26
Published online: 2022-09-29

DOI: 10.5802/crmath.371

Classification: 11A15, 11A07, 11R11

Author's affiliations:

Qing-Hu Hou ¹; Hao Pan ²; Zhi-Wei Sun ³

¹ School of Mathematics, Tianjin University, Tianjin 300350, People’s Republic of China
² School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, People’s Republic of China
³ Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Qing-Hu Hou and Hao Pan and Zhi-Wei Sun},
     title = {A new theorem on quadratic residues modulo primes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1065--1069},
     year = {2022},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     doi = {10.5802/crmath.371},
     language = {en},
}

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Qing-Hu Hou; Hao Pan; Zhi-Wei Sun. A new theorem on quadratic residues modulo primes. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1065-1069. doi: 10.5802/crmath.371

References
Cited by

[1] Henri Cohen A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, 138, Springer, 1993 | DOI | Zbl

[2] Harold Davenport The Higher Arithmetic. An Introduction to the Theory of Numbers, Cambridge University Press, 2008

[3] Qing-Hu Hou; Zhi-Wei Sun Sequence A320159 at OEIS (On-Line Encyclopedia of Integer Sequences), 2018 (http://oeis.org/A320159)

[4] Kenneth Ireland; Michael Rosen A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer, 1990 | DOI

[5] Zhi-Wei Sun Quadratic residues and related permutations and identities, Finite Fields Appl., Volume 59 (2019), pp. 246-283 | Zbl | MR

[6] Zhi-Wei Sun Quadratic residues and quartic residues modulo primes, Int. J. Number Theory, Volume 16 (2020) no. 8, pp. 1833-1858 | MR | Zbl

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