Consider the equation x^2+y^5=z^3 where x, y, and z, are positive integers.

(A) Can you give at least three solutions to it?

(B) Determine whether or not there is an infinite number of solutions.

Something is not clear to me : in your demonstration, you have

...in conjunction with the identity

2^(k-1) + 2^(k-1) = 2^k ?? How ?