x^3 + y^3 + z^3 = 3xyz

[franktaw at netscape.net]

Leaving alpha out of it, this factorization is
 
x^3+y^3+z^3-3xyz = (x+y+z)*(x^2+y^2+z^2-xy-xz-yz).
 
A proof escapes me at the moment, but I'm sure the second term is >= 0 for all real x,y,z, with zeros only for x=y=z.  This implies that the two classes of solutions described below are the only integer solutions. 
 
Franklin T. Adams-Watters
 
 
-----Original Message-----
From: Edwin Clark eclark at math.usf.edu


On Mon, 24 Apr 2006, Emeric Deutsch wrote:

> Dear Seqfans,
> Do you know any literature on the solutions of
> x^3 + y^3 + z^3 = 3xyz in integers?
> Any other solutions than x=y=z=1,2,3,... ?

Since 

x^3+y^3+z^3-3*x*y*z = (x-z-z*alpha+y*alpha)*(x-y-y*alpha+z*alpha)*(z+x+y)

where alpha is a root of x^2 + x + 1,

any roots of x+y+z=0 will be a solution. ...
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