> For the complete documentation index, see [llms.txt](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/er-ci-tong-yu-shi-he-ping-fang-sheng-yu/mopping-fang-sheng-yu-pan-duan.md).

# 模p平方剩余判断

> * **METHOD1: 定理**
>
>   设$$p$$为奇素数
>
>   * $$\left(\frac{2}{p}\right)={\left(-1\right)}^{\frac{p^2-1}{8}}$$
>   * 若$$\left(a,2p\right)=1$$，则$$\left(\frac{a}{p}\right)={\left(-1\right)}^{T\_{\left(a,p\right)}}$$，其中$$T\_{\left(a,p\right)}=\sum\_\limits{k=1}^{\frac{p-1}{2}}{\left\[\frac{a\cdot k}{p}\right]}$$
> * **METHOD2: 二次互反律**
>
>   若$$p,q$$为互素奇素数，则$$\left(\frac{p}{q}\right)={\left(-1\right)}^{\frac{p-1}{2}\cdot\frac{q-1}{2}}\left(\frac{p}{q}\right)$$