Difference between revisions of "Euler's Four-Square Identity"
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Mathematically, for any eight [[Complex numbers|complex numbers]] <math>x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4</math>, we must have <cmath>(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2</cmath> <cmath>+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2</cmath> <cmath>+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2</cmath> <cmath>+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.</cmath> | Mathematically, for any eight [[Complex numbers|complex numbers]] <math>x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4</math>, we must have <cmath>(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2</cmath> <cmath>+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2</cmath> <cmath>+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2</cmath> <cmath>+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.</cmath> | ||
| + | |||
| + | This is '''Euler's Four-Square Identity'''. | ||
==Proof== | ==Proof== | ||
Revision as of 07:50, 29 March 2019
Identity
The product of the sum of the four squares is itself, the sum of four squares.
Mathematically, for any eight complex numbers 
, we must have ![\[(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)\]](http://latex.artofproblemsolving.com/6/b/6/6b6fc940cccd9e26036d3f56a9b7842087b693e3.png)
![\[=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2\]](http://latex.artofproblemsolving.com/7/d/3/7d3e8fc99dee75a54c48f5ab4044d84c7bdf9180.png)
![\[+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2\]](http://latex.artofproblemsolving.com/d/3/f/d3fe8021c88eaea0a2d2df6f02d536fea296da26.png)
![\[+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2\]](http://latex.artofproblemsolving.com/e/4/8/e48a7385a25ce27a5c9bf7c8f2c455ea15429307.png)
![\[+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.\]](http://latex.artofproblemsolving.com/2/6/8/268b1ee217541ac04dc2ef23023e64b2659a20c8.png)
This is Euler's Four-Square Identity.
