Determinants concerning Legendre symbols
Hai-Liang Wu · Comptes Rendus Mathématique · 2021
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 ≡ 1 ( mod 4 ) . Then for the determinant S ( 1 , p ) : = det i 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.