Article
| Title: | Linear congruences and a conjecture of Bibak (English) |
| Author: | Babu, Chinnakonda Gnanamoorthy Karthick |
| Author: | Bera, Ranjan |
| Author: | Sury, Balasubramanian |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 4 |
| Year: | 2024 |
| Pages: | 1185-1206 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences $\sum _{i=1}^k a_i x_i \equiv b \pmod n$. 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. (English) |
| Keyword: | system of congruence |
| Keyword: | restricted linear congruence |
| Keyword: | Ramanujan sum |
| Keyword: | discrete Fourier transform |
| MSC: | 11A25 |
| MSC: | 11D79 |
| MSC: | 11P83 |
| MSC: | 11T24 |
| MSC: | 11T55 |
| DOI: | 10.21136/CMJ.2024.0151-24 |
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| Date available: | 2024-12-15T06:39:17Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152696 |
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