四平方定理
t 1 2 + t 2 2 + t 3 2 + t 4 2 = e f ≤ 4 ( f − 1 2 ) 2 < f 2 {\displaystyle t_{1}^{2}+t_{2}^{2}+t_{3}^{2}+t_{4}^{2}=ef\leq {4\left({\frac {f-1}{2}}\right)^{2}}<f^{2}}
もしも t 1 = t 2 = t 3 = t 4 = 0 {\displaystyle {t_{1}}={t_{2}}={t_{3}}={t_{4}}=0} ならば s 1 ≡ s 2 ≡ s 3 ≡ s 4 ≡ 0 ( mod f ) {\displaystyle {s_{1}}\equiv {s_{2}}\equiv {s_{3}}\equiv {s_{4}}\equiv 0\;({\bmod {\;}}f)} であるから s 1 2 + s 2 2 + s 3 2 + s 4 2 = f p ( mod f 2 ) {\displaystyle s_{1}^{2}+s_{2}^{2}+s_{3}^{2}+s_{4}^{2}=fp\;({\bmod {\;}}f^{2})} である。これは p {\displaystyle p} が素数であるという仮定に背くから e ≥ 1 {\displaystyle {e}\geq 1} である。四平方恒等式により
( f p ) ( e f ) = ( s 1 2 + s 2 2 + s 3 2 + s 4 2 ) ( t 1 2 + t 2 2 + t 3 2 + t 4 2 ) = ( s 1 t 1 + s 2 t 2 + s 3 t 3 + s 4 t 4 ) 2 + ( s 1 t 2 − s 2 t 1 + s 3 t 4 − s 4 t 3 ) 2 + ( s 1 t 3 − s 2 t 4 − s 3 t 1 + s 4 t 2 ) 2 + ( s 1 t 4 + s 2 t 3 − s 3 t 2 − s 4 t 1 ) 2 {\displaystyle {\begin{aligned}(fp)(ef)&=(s_{1}^{2}+s_{2}^{2}+s_{3}^{2}+s_{4}^{2})(t_{1}^{2}+t_{2}^{2}+t_{3}^{2}+t_{4}^{2})\\&=(s_{1}t_{1}+s_{2}t_{2}+s_{3}t_{3}+s_{4}t_{4})^{2}+(s_{1}t_{2}-s_{2}t_{1}+s_{3}t_{4}-s_{4}t_{3})^{2}+(s_{1}t_{3}-s_{2}t_{4}-s_{3}t_{1}+s_{4}t_{2})^{2}+(s_{1}t_{4}+s_{2}t_{3}-s_{3}t_{2}-s_{4}t_{1})^{2}\end{aligned}}}
s i ≡ t i ( mod f ) {\displaystyle {s_{i}}\equiv {t_{i}}\;({\bmod {\;}}f)} であるから s 1 t 1 + s 2 t 2 + s 3 t 3 + s 4 t 4 ≡ s 1 2 + s 2 2 + s 3 2 + s 4 2 ≡ 0 ( mod f ) {\displaystyle {s_{1}t_{1}+s_{2}t_{2}+s_{3}t_{3}+s_{4}t_{4}}\equiv {s_{1}^{2}+s_{2}^{2}+s_{3}^{2}+s_{4}^{2}}\equiv {0}\;({\bmod {\;}}f)} であり、他の項も同様であるから
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。