Combinatorics, Number theory
Determinants concerning Legendre symbols
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 651-655.

The evaluations of determinants with Legendre symbol entries have close relation with combinatorics and character sums over finite fields. Recently, Sun [9] posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result:

Let $p={a}^{2}+4{b}^{2}$ be a prime with $a,b$ integers and $a\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$. Then for the determinant

 $S\left(1,p\right):=det{\left[\left(\frac{{i}^{2}+{j}^{2}}{p}\right)\right]}_{1\le i,j\le \frac{p-1}{2}},$

the number $S\left(1,p\right)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.205
Classification: 11C20,  11L10,  11R18
Hai-Liang Wu 1

1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China
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Hai-Liang Wu. Determinants concerning Legendre symbols. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 651-655. doi : 10.5802/crmath.205. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.205/

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