Difference between revisions of "Euler's Four-Square Identity"
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==Proof== | ==Proof== | ||
| − | First, let us expand the left-hand side of the identity: <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) = x_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.</cmath> Thus, we 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) | + | First, let us expand the left-hand side of the identity: <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) = x_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.</cmath> Thus, we 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) = x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2</cmath> <cmath>+ x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2</cmath> <cmath>+ x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2</cmath> <cmath>+ x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2. \text{ (1)}</cmath> |
Now, | Now, | ||
Revision as of 08:08, 29 March 2019
Identity
The Four-Square Identity, credited to Leonhard Euler, states that 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)
Proof
First, let us expand the left-hand side of the identity: ![\[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)\]](http://latex.artofproblemsolving.com/7/7/e/77e8c69805418fe82050c4c2f22ff27ac65b4fe8.png)
![\[= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.\]](http://latex.artofproblemsolving.com/b/5/0/b507c2728a218499131d4083713b3ef8d0dc48ed.png)
Thus, we have ![\[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2\]](http://latex.artofproblemsolving.com/d/b/2/db2c6de8c4286f0edaa413d6866281078f5b27db.png)
![\[+ x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2\]](http://latex.artofproblemsolving.com/b/3/4/b34721ae634891a90a39c0adae5f988262377d9a.png)
![\[+ x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2\]](http://latex.artofproblemsolving.com/8/c/6/8c693a66ac907feeb00ac7bdb4300cbe0749f895.png)
![\[+ x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2. \text{ (1)}\]](http://latex.artofproblemsolving.com/0/1/9/019ce590756e918d5f8e35cbceafd7b700b9bcab.png)
Now,
