Exponential growth

\[\large p^x - y^3 =1\]

Solve the equation above over integral values of \(x,y,p\).

Details and assumptions:

  • Input the answer as the sum of possible answers. As an explicit example, input 12 if the only triplet satisfying the equation is \((1,1,10)\) .

  • \(p\) is a prime.

  • \(x,y\) are positive integers.

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