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

Authors : Renz Jimwel MİNA, Jerico BACANİ

Pages : 55-58

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Publication Date : 2022-04-15

Article Type : Research Paper

Abstract :The exponential Diophantine equations of type p x + q y = z 2 px+qy=z2 have been widely studied over the past decade. Authors studied these equations by considering primes p p and q q , and in general, for positive integers p p and q q . In this paper, we will be extending the study to Diophantine equations of type p x + q y = z 3 . px+qy=z3. In particular, we will be working with Diophantine equations of type p x + insert ignore into journalissuearticles values( p + 4 ); y = z 3 , px+insert ignore into journalissuearticles values(p+4);y=z3, where p p and p + 4 p+4 are cousin primes; that is, primes that differ by four. We state some sufficient conditions for the non-existence of solutions of equation insert ignore into journalissuearticles values(1); insert ignore into journalissuearticles values(???); 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.
Keywords : Diophantine equation, exponential Diophantine equation, Jacobi symbol, quadratic reciprocity, rational cubic residues