# Cubic Cube?

Number Theory Level 5

Find the triplet of positive integers that satisfy

$${ x }^{ 3 }={ 3 }^{ y }\cdot { 7 }^{ z }+8$$

Submit your answer as $$x+y+z$$.

Note

It's easy to try some values for $$x$$, $$y$$ and $$z$$, and find out the integers that satisfy the equation... to show that is the unique triplet of positive integers, can be more complicated. There's a short, relatively, solution that makes use of Pell's equation; also, I've found one appyling basic modular arithmetic.

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