Comptes Rendus
Combinatorics, Number theory
Determinants concerning Legendre symbols
Hai-Liang Wu1
1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China
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 +4b 2 be a prime with a,b integers and a≡1(mod4). Then for the determinant

S(1,p):=deti 2 +j 2 p 1≤i,j≤p-1 2 ,

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

Received:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Determinants concerning {Legendre} symbols},
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     doi = {10.5802/crmath.205},
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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

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