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From: alainverghote on 3 Jun 2010 04:46
Good morning,

It sems we could consider equalities modulo
a given power.
For degree 2 , Ex 2(3t+2x)^2<=>(4t+3x)^2<=>24xt
leading to
x^2 +2(3t+2x)^2 = (4t+3x)^2 +2t^2
x^2 + 2y^2 = z^2 + 2t^2 a quadratic form

For degree 3
Ex (x+y+z)^3 <=>3(x+y)(y+z)(z+x) modulo cubes x^3,y^3,z^3 (1)
Parametrizing x=-2a+b+c,y=a-2b+c,z=a+b-2c or (x+y+z) = 0
Gives:
(-2a+b+c)^3+(a-2b+c)^3+(a+b-2c)^3-3(-2a+b+c)(a-2b+c)(a+b-2c)=0
Namely a particular solution to the known equation
x^3+y^3+z^3-3xyz = 0 (2)

Example, case {a,b,c} = {7,11,13}
10^3+(-2)^3+(-8)^3 - 3*10*(-2)*(-8) = 0

Do you know more general solutions to (2) ?

Best regards,
Alain


From: Robert Israel on 3 Jun 2010 13:36
"alainverghote(a)gmail.com" <alainverghote(a)gmail.com> writes:

> Good morning,
>
> It sems we could consider equalities modulo
> a given power.
> For degree 2 , Ex 2(3t+2x)^2<=>(4t+3x)^2<=>24xt
> leading to
> x^2 +2(3t+2x)^2 = (4t+3x)^2 +2t^2
> x^2 + 2y^2 = z^2 + 2t^2 a quadratic form
>
> For degree 3
> Ex (x+y+z)^3 <=>3(x+y)(y+z)(z+x) modulo cubes x^3,y^3,z^3 (1)
> Parametrizing x=-2a+b+c,y=a-2b+c,z=a+b-2c or (x+y+z) = 0
> Gives:
> (-2a+b+c)^3+(a-2b+c)^3+(a+b-2c)^3-3(-2a+b+c)(a-2b+c)(a+b-2c)=0
> Namely a particular solution to the known equation
> x^3+y^3+z^3-3xyz = 0 (2)
>
> Example, case {a,b,c} = {7,11,13}
> 10^3+(-2)^3+(-8)^3 - 3*10*(-2)*(-8) = 0
>
> Do you know more general solutions to (2) ?
>
> Best regards,
> Alain

x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz)

so of course anything with x+y+z = 0 is a solution. The other
factor x^2+y^2+z^2-xy-xz-yz = (y - x/2 - z/2)^2 + (3/4) (z-x)^2,
so for real x,y,z it is never 0 unless x=y=z.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: alainverghote on 4 Jun 2010 06:33
Dear Robert,

Thanks for answering.
x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz)

This factorization also works for more variables:
Example
x^3+y^3+z^3+t^3-3(xyz+yzt+xzt+xyt) =
(x+y+z+t)(x^2+y^2+z^2+t^2-xy-yt-yz-zt-xt-yz)
...........

Alain
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