# One to rule three cubes

Number Theory Level 5

Let the equation $$1 = (1+9m^3)^3 + (9m^4)^3 + (-9m^4-mx)^3$$ have solutions for all $$m \in \mathbb{N}$$ and a certain $$x \in \mathbb{N}$$.

What is the smallest positive integer $$x$$ for this to be true?

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