The Distribution Properties of Consecutive Quadratic Residue Sequences
We consider any prime number p. Let k,s be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference s and length k, where the sequence entries are from the set of quadratic residue modulo p or the set of quadratic nonresidue modulo p. The numbers...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/5253261 |
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| _version_ | 1849308566529245184 |
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| author | Xiao Wang Hao Fang |
| author_facet | Xiao Wang Hao Fang |
| author_sort | Xiao Wang |
| collection | DOAJ |
| description | We consider any prime number p. Let k,s be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference s and length k, where the sequence entries are from the set of quadratic residue modulo p or the set of quadratic nonresidue modulo p. The numbers of such sequences are denoted as Npk,s and Np′k,s, respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo p and character sums, to study the numbers Npk,s and Np′k,s. Exact formulas are given for certain values of k and s under some restrictions. In addition, estimation formulas in other cases are given. |
| format | Article |
| id | doaj-art-043cb24122c54fb49a4adb043bdd282c |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-043cb24122c54fb49a4adb043bdd282c2025-08-20T03:54:25ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/5253261The Distribution Properties of Consecutive Quadratic Residue SequencesXiao Wang0Hao Fang1School of ScienceSchool of MathematicsWe consider any prime number p. Let k,s be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference s and length k, where the sequence entries are from the set of quadratic residue modulo p or the set of quadratic nonresidue modulo p. The numbers of such sequences are denoted as Npk,s and Np′k,s, respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo p and character sums, to study the numbers Npk,s and Np′k,s. Exact formulas are given for certain values of k and s under some restrictions. In addition, estimation formulas in other cases are given.http://dx.doi.org/10.1155/2023/5253261 |
| spellingShingle | Xiao Wang Hao Fang The Distribution Properties of Consecutive Quadratic Residue Sequences Journal of Mathematics |
| title | The Distribution Properties of Consecutive Quadratic Residue Sequences |
| title_full | The Distribution Properties of Consecutive Quadratic Residue Sequences |
| title_fullStr | The Distribution Properties of Consecutive Quadratic Residue Sequences |
| title_full_unstemmed | The Distribution Properties of Consecutive Quadratic Residue Sequences |
| title_short | The Distribution Properties of Consecutive Quadratic Residue Sequences |
| title_sort | distribution properties of consecutive quadratic residue sequences |
| url | http://dx.doi.org/10.1155/2023/5253261 |
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