二個の平方数の和
( − 2 8 n + 1 ) = ( − 1 8 n + 1 ) ( 2 8 n + 1 ) = ( − 1 ) 1 − 1 2 ( − 1 ) 1 − 1 8 = 1 , ( − 2 8 n + 3 ) = ( − 1 8 n + 3 ) ( 2 8 n + 3 ) = ( − 1 ) 3 − 1 2 ( − 1 ) 9 − 1 8 = 1 , ( − 2 8 n + 5 ) = ( − 1 8 n + 5 ) ( 2 8 n + 5 ) = ( − 1 ) 5 − 1 2 ( − 1 ) 25 − 1 8 = − 1 , ( − 2 8 n + 7 ) = ( − 1 8 n + 7 ) ( 2 8 n + 7 ) = ( − 1 ) 7 − 1 2 ( − 1 ) 49 − 1 8 = − 1 {\displaystyle {\begin{aligned}&\left({\frac {-2}{8n+1}}\right)=\left({\frac {-1}{8n+1}}\right)\left({\frac {2}{8n+1}}\right)=(-1)^{\frac {1-1}{2}}(-1)^{\frac {1-1}{8}}=1,\\&\left({\frac {-2}{8n+3}}\right)=\left({\frac {-1}{8n+3}}\right)\left({\frac {2}{8n+3}}\right)=(-1)^{\frac {3-1}{2}}(-1)^{\frac {9-1}{8}}=1,\\&\left({\frac {-2}{8n+5}}\right)=\left({\frac {-1}{8n+5}}\right)\left({\frac {2}{8n+5}}\right)=(-1)^{\frac {5-1}{2}}(-1)^{\frac {25-1}{8}}=-1,\\&\left({\frac {-2}{8n+7}}\right)=\left({\frac {-1}{8n+7}}\right)\left({\frac {2}{8n+7}}\right)=(-1)^{\frac {7-1}{2}}(-1)^{\frac {49-1}{8}}=-1\\\end{aligned}}}