تقرير
Problems and results on determinants involving Legendre symbols
| العنوان: | Problems and results on determinants involving Legendre symbols |
|---|---|
| المؤلفون: | Sun, Zhi-Wei |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11A15, 11C20, 15A15 |
| الوصف: | In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. Comment: 24 pages. Refined version with typos corrected |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.03626 |
| رقم الأكسشن: | edsarx.2405.03626 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |