Linear Congruences and a Conjecture of Bibak
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909127117111296 |
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| author | Babu, C. G. Karthick Bera, Ranjan Sury, B. |
| author_facet | Babu, C. G. Karthick Bera, Ranjan Sury, B. |
| contents | We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions for the number of solutions where $x_i$'s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1 \geq \cdots \geq x_k$ or, with strict order restrictions $x_1> \cdots > x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01923 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Congruences and a Conjecture of Bibak Babu, C. G. Karthick Bera, Ranjan Sury, B. Number Theory 11D79, 11P83, 11A25, 11T55, 11T24 We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions for the number of solutions where $x_i$'s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1 \geq \cdots \geq x_k$ or, with strict order restrictions $x_1> \cdots > x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties. |
| title | Linear Congruences and a Conjecture of Bibak |
| topic | Number Theory 11D79, 11P83, 11A25, 11T55, 11T24 |
| url | https://arxiv.org/abs/2403.01923 |