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 \(x+y+z\).


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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