ソフィー・ジェルマンの恒等式
ソフィー・ジェルマンの恒等式
\[ a^{4}+4b^{4}=\left(a^{2}+2ab+2b^{2}\right)\left(a^{2}-2ab+2b^{2}\right) \]
\[ a^{4}+4b^{4}=\left(a^{2}+2ab+2b^{2}\right)\left(a^{2}-2ab+2b^{2}\right) \]
\begin{align*}
a^{4}+4b^{4} & =a^{4}+4a^{2}b^{2}+4b^{4}-4a^{2}b^{2}\\
& =\left(a^{2}+2b^{2}\right)^{2}-\left(2ab\right)^{2}\\
& =\left(a^{2}+2ab+2b^{2}\right)\left(a^{2}-2ab+2b^{2}\right)
\end{align*}
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ブラーマグプタ2平方恒等式
\[
\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac\pm bd\right)^{2}+\left(ad\mp bc\right)^{2}
\]
オイラーの4平方恒等式
\[
\left(a_{0}^{\;2}+a_{1}^{\;2}+a_{2}^{\;2}+a_{3}^{\;2}\right)\left(b_{0}^{\;2}+b_{1}^{\;2}+b_{2}^{\;2}+b_{3}^{\;2}\right)=\left(a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}\right)^{2}+\left(a_{0}b_{1}+a_{1}b_{0}+a_{2}b_{3}-a_{3}b_{2}\right)^{2}+\left(a_{0}b_{2}-a_{1}b_{3}+a_{2}b_{0}+a_{3}b_{1}\right)^{2}+\left(a_{0}b_{3}+a_{1}b_{2}-a_{2}b_{1}+a_{3}b_{0}\right)^{2}
\]
2次式の実数の範囲で因数分解
\[
a^{2}\pm2ab+b^{2}=\left(a\pm b\right)^{2}
\]
n乗根の因数分解
\[
z^{n}-1=\prod_{k=1}^{n}\left(z-e^{\frac{2\pi}{n}ki}\right)
\]