A new theorem on quadratic residues modulo primes

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

doi:10.5802/crmath.371

Théorie des nombres

A new theorem on quadratic residues modulo primes

Hou, Qing-Hu ¹ ; Pan, Hao ² ; Sun, Zhi-Wei 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, Tome 360 (2022) no. G9, pp. 1065-1069.

Résumé

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 $p$ .

Reçu le : 2022-03-08
Accepté le : 2022-04-26
Publié le : 2022-09-29

DOI : 10.5802/crmath.371

Classification : 11A15, 11A07, 11R11

Affiliations des auteurs :

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

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     title = {A new theorem on quadratic residues modulo primes},
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     pages = {1065--1069},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.371/}
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Hou, Qing-Hu; Pan, Hao; Sun, Zhi-Wei. A new theorem on quadratic residues modulo primes. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1065-1069. doi : 10.5802/crmath.371. http://www.numdam.org/articles/10.5802/crmath.371/

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

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

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

[6] Sun, Zhi-Wei 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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