sum of 8 squares

**simplependulum** · August 7th 2009, 11:02 PM

If $X$ and $Y$ can be written as the sum of 8 squares , then their product is also the sum of 8 squares .

that means

$( a^2 + b^2 + c^2 + d^2 + e^2 + f^2 +g^2 + h^2)( A^2 + B^2 + C^2 + D^2 + E^2 + F^2 + G^2 +H^2) =$

$\alpha^2 + \beta^2 + \gamma^2 + \delta^2 + \epsilon^2 + \zeta^2 + \eta^2 + \theta^2$

Show that the relationship between $a,b,c,d...A,B,C,D...\alpha , \beta , \gamma , \delta ...$ is :

$\left(\begin{array}{cccccccc}a&b&c&d&e&f&g&-h\\-b&a&-d&c&-f&e&h&g\\c&-d&-a&b&g&h&-e&f\\d&c&-b&-a&h&-g&f&e\\-e&f&g&h&a&-b&-c&d\\-f&-e&h&-g&b&a&d&c\\g&h&e&-f&-c&d&-a&b\\h&-g&f&e&-d&-c&b&a\end{array}\right)$ $\left( \begin{array}{c}A\\B\\C\\D\\E\\F\\G\\H\end{array}\ right) = \left(\begin{array}{c}\alpha\\ \beta\\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \eta \\ \theta\end{array}\right)$

**NonCommAlg** · August 9th 2009, 09:12 AM

Originally Posted by simplependulum

If $X$ and $Y$ can be written as the sum of 8 squares , then their product is also the sum of 8 squares .

that means

$( a^2 + b^2 + c^2 + d^2 + e^2 + f^2 +g^2 + h^2)( A^2 + B^2 + C^2 + D^2 + E^2 + F^2 + G^2 +H^2) =$

$\alpha^2 + \beta^2 + \gamma^2 + \delta^2 + \epsilon^2 + \zeta^2 + \eta^2 + \theta^2$

Show that the relationship between $a,b,c,d...A,B,C,D...\alpha , \beta , \gamma , \delta ...$ is :

$\left(\begin{array}{cccccccc}a&b&c&d&e&f&g&-h\\-b&a&-d&c&-f&e&h&g\\c&-d&-a&b&g&h&-e&f\\d&c&-b&-a&h&-g&f&e\\-e&f&g&h&a&-b&-c&d\\-f&-e&h&-g&b&a&d&c\\g&h&e&-f&-c&d&-a&b\\h&-g&f&e&-d&-c&b&a\end{array}\right)$ $\left( \begin{array}{c}A\\B\\C\\D\\E\\F\\G\\H\end{array}\ right) = \left(\begin{array}{c}\alpha\\ \beta\\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \eta \\ \theta\end{array}\right)$

it's straightforward if you're familiar with $\mathbb{O},$ the normed division algebra of octonions. your question then is equivalent to proving that for any $x,y \in \mathbb{O}: \ ||x|| \cdot ||y|| = ||xy||.$

**luobo** · September 4th 2009, 04:37 PM

This problem is related to the Degen's eight-square identity.

Degen's eight-square identity - Wikipedia, the free encyclopedia

**ThePerfectHacker** · September 4th 2009, 05:10 PM

You can also do this with 16 and 32 squares by generalizing the octions. Not sure about 64 and beyond.

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