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Math Help - sum of 8 squares

  1. #1
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    sum of 8 squares

    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)
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  2. #2
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    Quote Originally Posted by simplependulum View Post
    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||.
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  3. #3
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    This problem is related to the Degen's eight-square identity.


    Degen's eight-square identity - Wikipedia, the free encyclopedia
    Last edited by mr fantastic; September 18th 2009 at 08:20 AM. Reason: Restored original reply
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  4. #4
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    You can also do this with 16 and 32 squares by generalizing the octions. Not sure about 64 and beyond.
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