Theorem 1 The Jacobi symbol has the following properties:
(i) $\left(\frac{1}{P}\right)=1$; when $(d, P)>1$, $\left(\frac{d}{P}\right)=0$; when $(d, P)=1$, $\left(\frac{d}{P}\right)$ takes values $\pm 1$.
(ii) $\left(\frac{d}{P}\right)=\left(\frac{d+P}{P}\right)$.
(iii) $\left(\frac{d c}{P}\right)=\left(\frac{d}{P}\right)\left(\frac{c}{P}\right)$.
(iv) $\left(\frac{d}{P_{1} P_{2}}\right)=\left(\frac{d}{P_{1}}\right)\left(\frac{d}{P_{2}}\right)$.
(v) When $(P, d)=1$, $\left(\frac{d^{2}}{P}\right)=\left(\frac{d}{P^{2}}\right)=1$.