  The evaluations of determinants with Legendre symbol entries have close
relation with character sums over finite fields. Recently, Sun 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 $a\equiv1\pmod4$. Then
for the determinant $$S(1,p):={\rm
det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$
the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed
by Cohen, Sun and Vsemirnov.

Number Theory