Number theory
A new theorem on quadratic residues modulo primes
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1065-1069.

Let $p>3$ be a prime, and let $\left(\frac{·}{p}\right)$ be the Legendre symbol. Let $b\in ℤ$ and $\epsilon \in \left\{±1\right\}$. We mainly prove that

 $\left|\left\{{N}_{p}\left(a,b\right):\phantom{\rule{4pt}{0ex}}1

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

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DOI: 10.5802/crmath.371
Classification: 11A15, 11A07, 11R11
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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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.371/

[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 | MR | Zbl

[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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