ヤコビ記号
{\displaystyle \left({\frac {ab}{n}}\right)=\left({\frac {a}{n}}\right)\left({\frac {b}{n}}\right),\quad {\text{so }}\left({\frac {a^{2}}{n}}\right)=\left({\frac {a}{n}}\right)^{2}=1{\text{ or }}0.}
5. ( a m n ) = ( a m ) ( a n ) , so ( a n 2 ) = ( a n ) 2 = 1 or 0. {\displaystyle \left({\frac {a}{mn}}\right)=\left({\frac {a}{m}}\right)\left({\frac {a}{n}}\right),\quad {\text{so }}\left({\frac {a}{n^{2}}}\right)=\left({\frac {a}{n}}\right)^{2}=1{\text{ or }}0.}
平方剰余の法則: m と n が奇数で互いに素な正整数である場合、
6. ( m n ) ( n m ) = ( − 1 ) m − 1 2 ⋅ n − 1 2 = { 1 if n ≡ 1 ( mod 4 ) or m ≡ 1 ( mod 4 ) , − 1 if n ≡ m ≡ 3 ( mod 4 ) {\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{{\tfrac {m-1}{2}}\cdot {\tfrac {n-1}{2}}}={\begin{cases}1&{\text{if }}n\equiv 1{\pmod {4}}{\text{ or }}m\equiv 1{\pmod {4}},\\-1&{\text{if }}n\equiv m\equiv 3{\pmod {4}}\end{cases}}}