ヤコビの四平方定理
12 = ( ± 2 ) 2 + ( ± 2 ) 2 + ( ± 2 ) 2 + 0 2 = ( ± 2 ) 2 + ( ± 2 ) 2 + 0 2 + ( ± 2 ) 2 = ( ± 2 ) 2 + 0 2 + ( ± 2 ) 2 + ( ± 2 ) 2 = 0 2 + ( ± 2 ) 2 + ( ± 2 ) 2 + ( ± 2 ) 2 = ( ± 3 ) 2 + ( ± 1 ) 2 + ( ± 1 ) 2 + ( ± 1 ) 2 = ( ± 1 ) 2 + ( ± 3 ) 2 + ( ± 1 ) 2 + ( ± 1 ) 2 = ( ± 1 ) 2 + ( ± 1 ) 2 + ( ± 3 ) 2 + ( ± 1 ) 2 = ( ± 1 ) 2 + ( ± 1 ) 2 + ( ± 1 ) 2 + ( ± 3 ) 2 {\displaystyle {\begin{aligned}12&=(\pm 2)^{2}+(\pm 2)^{2}+(\pm 2)^{2}+0^{2}\\&=(\pm 2)^{2}+(\pm 2)^{2}+0^{2}+(\pm 2)^{2}\\&=(\pm 2)^{2}+0^{2}+(\pm 2)^{2}+(\pm 2)^{2}\\&=0^{2}+(\pm 2)^{2}+(\pm 2)^{2}+(\pm 2)^{2}\\&=(\pm 3)^{2}+(\pm 1)^{2}+(\pm 1)^{2}+(\pm 1)^{2}\\&=(\pm 1)^{2}+(\pm 3)^{2}+(\pm 1)^{2}+(\pm 1)^{2}\\&=(\pm 1)^{2}+(\pm 1)^{2}+(\pm 3)^{2}+(\pm 1)^{2}\\&=(\pm 1)^{2}+(\pm 1)^{2}+(\pm 1)^{2}+(\pm 3)^{2}\\\end{aligned}}}
であり、符号と順序を区別すれば96個になる。