# 二次剩余

## **定义**

$$m$$为正整数，若$$x^2\equiv a\left(\mod m\right),\left(a,m\right)=1$$有解，则$$a$$为$$m$$的二次剩余，否则为二次非剩余

## **欧拉判别条件**

 $$p$$为奇素数，$$\left(a,p\right)=1$$

* $$a$$是模$$p$$的平方剩余$$\Longleftrightarrow a^{\frac{p-1}{2}}\equiv1\left(\mod p\right)$$
* $$a$$是模$$p$$的非平方剩余$$\Longleftrightarrow a^{\frac{p-1}{2}}\equiv-1\left(\mod p\right)$$

**推论:** 

* $$p$$为奇素数，$$\left(a\_1,p\right)=1,\left(a\_2,p\right)=1$$，则$$a\_1\cdot a\_2$$为模$$p$$的平方非剩余$$\Longleftrightarrow a\_1,a\_2$$同为模$$p$$的平方剩余或平方非剩余
* 平方剩余与平方非剩余数量相等


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