Gaussian integer solutions of the Diophantine equation x4 + y4 = z3 for x 6= y / Shahrina Ismail, Kamel Ariffin Mohd Atan and Diego Sejas Viscarra

The investigation of determining solutions for the Diophantine equation x4 + y4 = z3 over the Gaussian integer field, for the specific case of x 6= y, is discussed. The discussion includes various preliminary results needed to build the future resolvent theory of the Diophantine equation studied. Our findings show the existence on infinitely many solutions. Since the analytical method used is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.