$ax^2 + by^2 = c$有有理点$⇔-bc,-ca,-ab$分别为mod $|a|,|b|,|c|$的二次剩余

hbghlyj

jstor.org/stable/2691462 写道：

$ax^2 + by^2 = c$, where $a, b, c \in \mathbb{Z}$, not all of the same sign, and $abc$ is square-free, has rational points if and only if $-bc, -ca, -ab$ are quadratic residues modulo $|a|, |b|,|c|$ respectively. This is Legendre's Criterion.

整数$a,b,c$不全同号，$abc$无平方因数，
$ax^2 + by^2 = c$有有理点$⇔-bc,-ca,-ab$分别为mod $|a|,|b|,|c|$的二次剩余。
怎么证明呢？