Article
| Title: | The tangent function and power residues modulo primes (English) |
| Author: | Sun, Zhi-Wei |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 971-978 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1\pmod {2m}$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod _{k\in R_m(p)}(1+\tan (\pi ak/p))$, where $$ R_m(p)=\{0<k<p\colon k\in \mathbb Z\ \text {is an}\ m\text {th power residue modulo}\ p\}. $$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb Z$, then $$ \prod _{k\in R_4(p)} \Big (1+\tan \pi \frac {ak}p\Big )=(-1)^{y}(-2)^{(p-1)/8}. $$ (English) |
| Keyword: | power residues modulo prime |
| Keyword: | the tangent function |
| Keyword: | identity |
| MSC: | 05A19 |
| MSC: | 11A15 |
| MSC: | 33B10 |
| idZBL: | Zbl 07729549 |
| idMR: | MR4632869 |
| DOI: | 10.21136/CMJ.2023.0395-22 |
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| Date available: | 2023-08-11T14:31:16Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151786 |
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| Reference: | [1] Berndt, B. C., Evans, R. J., Williams, K. S.: Gauss and Jacobi Sums.Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York (1998). Zbl 0906.11001, MR 1625181 |
| Reference: | [2] Cox, D. A.: Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication.Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. John Wiley & Sons, New York (1989). Zbl 0956.11500, MR 1028322, 10.1002/9781118400722 |
| Reference: | [3] Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory.Graduate Texts in Mathematics 84. Springer, New York (1990). Zbl 0712.11001, MR 1070716, 10.1007/978-1-4757-2103-4 |
| Reference: | [4] Sun, Z.-W.: Trigonometric identities and quadratic residues.Publ. Math. Debr. 102 (2023), 111-138. Zbl 7650970, MR 4556502, 10.5486/PMD.2023.9352 |
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