Gaussian Integer Solutions of the Diophantine Equation x4+y4=z3 for xy

The investigation of determining solutions for the Diophantine equation x4 + y4 = z3 over the Gaussian integer ring for the specific case of x * y is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.