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proof of Lagrange's four-square theorem

Major Section: 
Reference
Type of Math Object: 
Proof

Mathematics Subject Classification

11P05 no label found

Comments

I do not understand how if
2m=x^2+y^2,
then
m=((x-y)/2)^2+((x+y)/2)^2.
Does anyone want to explain?

Well, just calculate it:

((x-y)/2)^2+((x+y)/2)^2=(x^2-2xy+y^2)/4+(x^2+2xy+y^2)/4
=(2x^2+2y^2)/4=(x^2+y^2)/2=2m/2=m

Hope this helps you.

--
"Do not meddle in the affairs of wizards for they are subtle and quick to anger."

If you expand the lower expression you get

(x^2 -2xy +y^2 +x^2 +2xy +y^2) / 4

= (2x^2 +2y^2) / 4

= (x^2 + y^2) / 2 (dividing by 2)

= 2m / 2 (see upper expression)

= m

as required

yours professor pineapple

2m = x^2 + y^2
m = (x^2 + y^2) / 2
m = (x^2 + 2xy + y^2 + x^2 - 2xy + y^2) / 4
m = (x^2 + 2xy + y^2) / 4 + (x^2 - 2xy + y^2) / 4
m = ((x + y)^2) / 2^2 + ((x - y)^2) / 2^2
m = ((x + y) / 2)^2 + ((x - y) / 2)^2

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