Evaluation of a determinant involving Legendre symbols
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916861791174656 |
|---|---|
| author | Ren, Chen-Kai Sun, Zhi-Wei |
| author_facet | Ren, Chen-Kai Sun, Zhi-Wei |
| contents | Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Evaluation of a determinant involving Legendre symbols Ren, Chen-Kai Sun, Zhi-Wei Number Theory 11C20, 15A15, 11A15 Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely. |
| title | Evaluation of a determinant involving Legendre symbols |
| topic | Number Theory 11C20, 15A15, 11A15 |
| url | https://arxiv.org/abs/2507.18589 |