On a Diophantine Equation of Type $p^x+q^y=z^3$

Abstract

The exponential Diophantine equations of type px+qy=z2px+qy=z2 have been widely studied over the past decade. Authors studied these equations by considering primes pp and qq, and in general, for positive integers pp and qq. In this paper, we will be extending the study to Diophantine equations of type px+qy=z3.px+qy=z3. In particular, we will be working with Diophantine equations of type px+(p+4)y=z3,px+(p+4)y=z3, where pp and p+4p+4 are cousin primes; that is, primes that differ by four. We state some sufficient conditions for the non-existence of solutions of equation (1)(???) on the set of positive integers. The proof uses some results in the theory of rational cubic residues as well as results in quadratic reciprocity, and some elementary techniques. It will be shown also that other Diophantine equations of similar type can also be studied with the approaches used in this paper.