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Define a primitive inverse-cubed triplet as any ordered triplet (x,y,z),(x,y,z),(x,y,z), where x,y,z∈Z+x,y,z\in\mathbb{Z}^+x,y,z∈Z+ and gcd(x,y,z)=1,\gcd{(x,y,z)}=1,gcd(x,y,z)=1, such that
1x3+1y3=1z3.\frac{1}{x^3}+\frac{1}{y^3}=\frac{1}{z^3}.x31+y31=z31.
How many ordered triplets (x,y,z)(x,y,z)(x,y,z) exist that are primitive inverse-cubed triplets?
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