On the integral solutions of the Diophantine equation x4 + y4 = 2kz3 where k > 1

Abstract

This paper is concerned with the existence, types, and the cardinality of the integral solutions of the Diophantine equation x4 + y4 = 2kz3, for k > 1. The objective of this paper is to develop methods to be used in finding all integer solutions to this equation. Results of the study show the existence of infinitely many integral solutions to this type of Diophantine equation for both cases, x = y and x ≠ y. For the case when x=y, the form of the solutions is given by (a, b, c) = (2k−1n3, 2k-1n3, 2k−1n4) when 1 ≤ k < 5, and (a, b, c) = (2k−1−3tn3, 2k−1−3tn3, 2k−1−4tn4), for t≤k−14 when k ≥ 5. Meanwhile, for the case when x ≠ y, the form of solutions is given by (a, b, c) = (2kun2, 2kvn2, 2kn3) or (a, b, c)= (2kdu, 2kdv, 2kdn), depending on the value of k. The main result obtained is a formulation of the generalized method to find all the solutions for this type of Diophantine equation.