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In this short article, it is established for the ti tle equation: (i) No solutions exist when p = 2. (ii) Exactly two solutions exist when p = 3. In both solutions q is prime. (iii) Exactly two solutions exist for each and eve ry prime p > 3 in which q is composite. Some numerical solutions are also exhib ited.

All the Solutions of the Diophantine Equation $p^3 + q^2 = z^3$