x^3 + y^3 + z^3 = 3xyz
Edwin Clark
eclark at math.usf.edu
Mon Apr 24 15:58:28 CEST 2006
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. For example things like,
n, -k, -(n-k)
PS I found the above factorization at
http://www.thiel.edu/mathproject/atps/chptr08/p072.htm
and verified it using Maple.
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