Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.

(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}) =

(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4})^{2} + (a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2} + (a_{1}b_{3}+a_{3}b_{1}+a_{4}b_{2}-a_{2}b_{4})^{2} + (a_{1}b_{4}+a_{4}b_{1}+a_{2}b_{3}-a_{3}b_{2})^{2}

It can be proved with elementary algebra or even by quaternions!

Further Reading:

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

[*11.62]

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This entry was posted on March 24, 2019 at 6:52 AM and is filed under fourfolds, Mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
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March 25, 2019 at 9:31 AM |

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