Factoring x^3+y^3+z^3: Conditions & Solutions

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Hey everyone first time poster here, I need help with some factoring of cubes. I know this might tie closely to Diophantine equations but here goes.

Under what condition is the expression x^3+y^3+z^3 factorable? Where x,y,z are positive whole numbers.
 
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well it seems to factor if x=1, y=2, z=3, and if x=y=z ≥ 2, and more generally if gcd(x,y,z) > 1.
 
My bad for not stating before. Let's assume the gcd(x,y,z) =1 and gcd(x,y)>1
 

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