A new theorem on quadratic residues modulo primes
Let $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $b\in \mathbb{Z}$ and $\varepsilon \in \lbrace \pm 1\rbrace $. We mainly prove that \[ \left|\left\lbrace N_p(a,b):\ 1\lbrace ax^2+b\rbrace _p$, and $\lbrace m\rbrace _p$ with $m\in \mathbb{Z}$ is the least nonnegative...
| Published in: | Comptes Rendus. Mathématique |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2022-09-01
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.371/ |
| _version_ | 1850088080645554176 |
|---|---|
| author | Hou, Qing-Hu Pan, Hao Sun, Zhi-Wei |
| author_facet | Hou, Qing-Hu Pan, Hao Sun, Zhi-Wei |
| author_sort | Hou, Qing-Hu |
| collection | DOAJ |
| container_title | Comptes Rendus. Mathématique |
| description | Let $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $b\in \mathbb{Z}$ and $\varepsilon \in \lbrace \pm 1\rbrace $. We mainly prove that
\[ \left|\left\lbrace N_p(a,b):\ 1\lbrace ax^2+b\rbrace _p$, and $\lbrace m\rbrace _p$ with $m\in \mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$. |
| format | Article |
| id | doaj-art-0c78ebbdb4e14668a47fe1fefba3b0ca |
| institution | Directory of Open Access Journals |
| issn | 1778-3569 |
| language | English |
| publishDate | 2022-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| spelling | doaj-art-0c78ebbdb4e14668a47fe1fefba3b0ca2025-08-20T00:10:05ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-09-01360G91065106910.5802/crmath.37110.5802/crmath.371A new theorem on quadratic residues modulo primesHou, Qing-Hu0Pan, Hao1Sun, Zhi-Wei2School of Mathematics, Tianjin University, Tianjin 300350, People’s Republic of ChinaSchool of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, People’s Republic of ChinaDepartment of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of ChinaLet $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $b\in \mathbb{Z}$ and $\varepsilon \in \lbrace \pm 1\rbrace $. We mainly prove that \[ \left|\left\lbrace N_p(a,b):\ 1\lbrace ax^2+b\rbrace _p$, and $\lbrace m\rbrace _p$ with $m\in \mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.371/ |
| spellingShingle | Hou, Qing-Hu Pan, Hao Sun, Zhi-Wei A new theorem on quadratic residues modulo primes |
| title | A new theorem on quadratic residues modulo primes |
| title_full | A new theorem on quadratic residues modulo primes |
| title_fullStr | A new theorem on quadratic residues modulo primes |
| title_full_unstemmed | A new theorem on quadratic residues modulo primes |
| title_short | A new theorem on quadratic residues modulo primes |
| title_sort | new theorem on quadratic residues modulo primes |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.371/ |
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