Comptes Rendus
Combinatoire, Théorie des nombres
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.

Reçu le :
Révisé le :
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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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Determinants concerning {Legendre} symbols},
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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/

[1] Bruce C. Berndt; Ronald J. Evans; Kenneth S. Williams Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1998 | Zbl

[2] Leonard Carlitz Some cyclotomic matrices, Acta Arith., Volume 5 (1959), pp. 293-308 | DOI | MR | Zbl

[3] Robin Chapman Determinants of Legendre symbol matrices, Acta Arith., Volume 115 (2004) no. 3, pp. 231-244 | DOI | MR | Zbl

[4] Robin Chapman My evil determinant problem (2012) (preprint, available at http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/evildet.pdf)

[5] Xiang-dong Hou Permutation polynomials over finite fields – a survey of recent advances, Finite Fields Appl., Volume 32 (2015), pp. 82-119 | MR | Zbl

[6] Kenneth Ireland; Michael Rosen A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer, 1990 | MR | Zbl

[7] Christian Krattenthaler Advanced determinant calculus, Sémin. Lothar. Comb. (1999), B42q, 67 pages | MR | Zbl

[8] Christian Krattenthaler Advanced determinant calculus: a complement, Linear Algebra Appl., Volume 411 (2005), pp. 68-166 | DOI | MR | Zbl

[9] Zhi-Wei Sun On some determinants with Legendre symbol entries, Finite Fields Appl., Volume 56 (2019), pp. 285-307 | MR | Zbl

[10] Zhi-Wei Sun Quadratic residues and related permutations and identities, Finite Fields Appl., Volume 59 (2019), pp. 246-283 | MR | Zbl

[11] Maxim Vsemirnov On the evaluation of R. Chapman’s “evil determinant”, Linear Algebra Appl., Volume 436 (2012) no. 11, pp. 4101-4106 | DOI | MR | Zbl

[12] Maxim Vsemirnov On R. Chapman’s “evil determinant”: case p≡1(mod4), Acta Arith., Volume 159 (2013) no. 4, pp. 331-344 | DOI | MR | Zbl

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