Euler’s Four-square Identity

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.

(a12+a22+a32+a42)(b12+b22+b32+b42) =

(a1b1-a2b2-a3b3-a4b4)2 + (a1b2+a2b1+a3b4-a4b3)2 + (a1b3+a3b1+a4b2-a2b4)2 + (a1b4+a4b1+a2b3-a3b2)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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One Response to “Euler’s Four-square Identity”

  1. Degen’s Eight-square Identity | Equivalent eXchange Says:

    […] the expressions above have an interesting symmetry, aside from the one on the upper left. Indeed, Euler’s Four-square Identity has a similar simpler symmetry. There is also a connection with Octonions if you are interested in […]

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