Degen's eight-square identity

In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: ( a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 + a 7 2 + a 8 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 + b 5 2 + b 6 2 + b 7 2 + b 8 2 ) = ( a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 − a 5 b 5 − a 6 b 6 − a 7 b 7 − a 8 b 8 ) 2 + ( a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 + a 5 b 6 − a 6 b 5 − a 7 b 8 + a 8 b 7 ) 2 + ( a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 + a 5 b 7 + a 6 b 8 − a 7 b 5 − a 8 b 6 ) 2 + ( a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 + a 5 b 8 − a 6 b 7 + a 7 b 6 − a 8 b 5 ) 2 + ( a 1 b 5 − a 2 b 6 − a 3 b 7 − a 4 b 8 + a 5 b 1 + a 6 b 2 + a 7 b 3 + a 8 b 4 ) 2 + ( a 1 b 6 + a 2 b 5 − a 3 b 8 + a 4 b 7 − a 5 b 2 + a 6 b 1 − a 7 b 4 + a 8 b 3 ) 2 + ( a 1 b 7 + a 2 b 8 + a 3 b 5 − a 4 b 6 − a 5 b 3 + a 6 b 4 + a 7 b 1 − a 8 b 2 ) 2 + ( a 1 b 8 − a 2 b 7 + a 3 b 6 + a 4 b 5 − a 5 b 4 − a 6 b 3 + a 7 b 2 + a 8 b 1 ) 2 {\displaystyle {\begin{aligned}&\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}\right)\left(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}+b_{5}^{2}+b_{6}^{2}+b_{7}^{2}+b_{8}^{2}\right)=\\[1ex]&\quad \left(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4}-a_{5}b_{5}-a_{6}b_{6}-a_{7}b_{7}-a_{8}b_{8}\right)^{2}+\\&\quad \left(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3}+a_{5}b_{6}-a_{6}b_{5}-a_{7}b_{8}+a_{8}b_{7}\right)^{2}+\\&\quad \left(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2}+a_{5}b_{7}+a_{6}b_{8}-a_{7}b_{5}-a_{8}b_{6}\right)^{2}+\\&\quad \left(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{8}-a_{6}b_{7}+a_{7}b_{6}-a_{8}b_{5}\right)^{2}+\\&\quad \left(a_{1}b_{5}-a_{2}b_{6}-a_{3}b_{7}-a_{4}b_{8}+a_{5}b_{1}+a_{6}b_{2}+a_{7}b_{3}+a_{8}b_{4}\right)^{2}+\\&\quad \left(a_{1}b_{6}+a_{2}b_{5}-a_{3}b_{8}+a_{4}b_{7}-a_{5}b_{2}+a_{6}b_{1}-a_{7}b_{4}+a_{8}b_{3}\right)^{2}+\\&\quad \left(a_{1}b_{7}+a_{2}b_{8}+a_{3}b_{5}-a_{4}b_{6}-a_{5}b_{3}+a_{6}b_{4}+a_{7}b_{1}-a_{8}b_{2}\right)^{2}+\\&\quad \left(a_{1}b_{8}-a_{2}b_{7}+a_{3}b_{6}+a_{4}b_{5}-a_{5}b_{4}-a_{6}b_{3}+a_{7}b_{2}+a_{8}b_{1}\right)^{2}\end{aligned}}} First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845).

Source: Wikipedia — Degen's eight-square identity (CC BY-SA 4.0)

Degen's eight-square identity

In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: ( a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 + a 7 2 + a 8 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 + b 5 2 + b 6 2 + b 7 2 + b 8 2 ) = ( a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 − a 5 b 5 − a 6 b 6 − a 7 b 7 − a 8 b 8 ) 2 + ( a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 + a 5 b 6 − a 6 b 5 − a 7 b 8 + a 8 b 7 ) 2 + ( a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 + a 5 b 7 + a 6 b 8 − a 7 b 5 − a 8 b 6 ) 2 + ( a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 + a 5 b 8 − a 6 b 7 + a 7 b 6 − a 8 b 5 ) 2 + ( a 1 b 5 − a 2 b 6 − a 3 b 7 − a 4 b 8 + a 5 b 1 + a 6 b 2 + a 7 b 3 + a 8 b 4 ) 2 + ( a 1 b 6 + a 2 b 5 − a 3 b 8 + a 4 b 7 − a 5 b 2 + a 6 b 1 − a 7 b 4 + a 8 b 3 ) 2 + ( a 1 b 7 + a 2 b 8 + a 3 b 5 − a 4 b 6 − a 5 b 3 + a 6 b 4 + a 7 b 1 − a 8 b 2 ) 2 + ( a 1 b 8 − a 2 b 7 + a 3 b 6 + a 4 b 5 − a 5 b 4 − a 6 b 3 + a 7 b 2 + a 8 b 1 ) 2 {\displaystyle {\begin{aligned}&\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}+a_{8}^{2}\right)\left(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}+b_{5}^{2}+b_{6}^{2}+b_{7}^{2}+b_{8}^{2}\right)=\\[1ex]&\quad \left(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4}-a_{5}b_{5}-a_{6}b_{6}-a_{7}b_{7}-a_{8}b_{8}\right)^{2}+\\&\quad \left(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3}+a_{5}b_{6}-a_{6}b_{5}-a_{7}b_{8}+a_{8}b_{7}\right)^{2}+\\&\quad \left(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2}+a_{5}b_{7}+a_{6}b_{8}-a_{7}b_{5}-a_{8}b_{6}\right)^{2}+\\&\quad \left(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{8}-a_{6}b_{7}+a_{7}b_{6}-a_{8}b_{5}\right)^{2}+\\&\quad \left(a_{1}b_{5}-a_{2}b_{6}-a_{3}b_{7}-a_{4}b_{8}+a_{5}b_{1}+a_{6}b_{2}+a_{7}b_{3}+a_{8}b_{4}\right)^{2}+\\&\quad \left(a_{1}b_{6}+a_{2}b_{5}-a_{3}b_{8}+a_{4}b_{7}-a_{5}b_{2}+a_{6}b_{1}-a_{7}b_{4}+a_{8}b_{3}\right)^{2}+\\&\quad \left(a_{1}b_{7}+a_{2}b_{8}+a_{3}b_{5}-a_{4}b_{6}-a_{5}b_{3}+a_{6}b_{4}+a_{7}b_{1}-a_{8}b_{2}\right)^{2}+\\&\quad \left(a_{1}b_{8}-a_{2}b_{7}+a_{3}b_{6}+a_{4}b_{5}-a_{5}b_{4}-a_{6}b_{3}+a_{7}b_{2}+a_{8}b_{1}\right)^{2}\end{aligned}}} First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845).

Source: Wikipedia "Degen's eight-square identity" · CC BY-SA 4.0

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