1. Determinants concerning Legendre symbols
- Author
-
Wu, Hai-Liang
- Subjects
Mathematics ,QA1-939 - Abstract
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 $a\equiv 1\pmod {4}$. Then for the determinant \[ S(1,p):=\det \Biggl [\Bigg (\frac{i^2+j^2}{p}\Bigg )\Biggr ]_{1\le i,j\le \frac{p-1}{2}}, \] the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
- Published
- 2021
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