At 01:12 PM 11/1/2003 +0000, you wrote:

>I expect you're right, Chris (if you correct your typo and say X = -

>r^n*Z ), and both the equation and your assertions have a familiar

>ring to them, but I can neither prove that no other solutions exist

>nor lay hands on a relevant reference.

>Help us out, please!

The wonderful text: L. J. Mordell "Diophantine Equations"

Academic Press, 1969, page 124-130 includes things like:

Theorem 3: Let a = p or p^2 where p = 2, 5 (mod 9) is a prime,

and let e be a unit in Q(r) (r^2 + r + 1 = 0). Then the equation

x^3 + y^3 + eaz^3 = 0

has no solutions (x,y,z,e) in Q(r) except

z = 0, x = -y, -ry, -r^2y

unless a = 2, when the equation also has the solutions

x^3 = y^3 = -e z^3, e = +/-1.

Proof is about one page and is done by choosing x, y, z

pairwise relatively prime and N(xyz) minimal. Then do

descent...

But we are wandering away from primes...

Chris