| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-08-11T14:31:16Z |
| Last updated:
|
2025-10-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151786 |
| . |
| 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 |
| . |
