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

Year 2022, Volume: 10 Issue: 1, 55 - 58, 15.04.2022

Renz Jimwel Mina Jerico Bacani

Abstract

The exponential Diophantine equations of type $p^x+q^y=z^2$ have been widely studied over the past decade. Authors studied these equations by considering primes $p$ and $q$, and in general, for positive integers $p$ and $q$. In this paper, we will be extending the study to Diophantine equations of type
$p^x+q^y=z^3.$
In particular, we will be working with Diophantine equations of type
$$p^x+(p+4{)}^y=z^3,$$
where $p$ and $p+4$ are cousin primes; that is, primes that differ by four. We state some sufficient conditions for the non-existence of solutions of equation $\text{(1)}$ 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.

Keywords

Diophantine equation, exponential Diophantine equation, Jacobi symbol, quadratic reciprocity, rational cubic residues

Supporting Institution

University of the Philippines Baguio

Thanks

The authors would like to thank the University of the Philippines Baguio for the support given in disseminating and publishing the results of the research study. The authors would also like to thank the referees for their time and effort to review the original manuscript, and give valuable comments and suggestions.

References

• [1] D. Acu, On a Diophantine equation 2x +5y = z2, Gen. Math. Vol:15, No.4 (2007), 145-148.
• [2] J. B. Bacani and J. F. T. Rabago, The complete set of solutions of the Diophantine equation px +qy = z2 for twin primes p and q, Int. J. Pure Appl. Math. Vol:104, No.4 (2015), 517-521.
• [3] Burton, D. M., Elementary Number Theory, Allyn and Bacon Inc. Boston, 1980.
• [4] Lemmermeyer, F., Reciprocity Laws from Euler to Eisenstein, Springer-Verlag Berlin, 2000.
• [5] J. F. T. Rabago, More on Diophantine equations of type px +qy = z2, Int. J. Math. Sci. Comp. Vol:3, No.1 (2013), 15-16.
• [6] J. F. T. Rabago, On an Open Problem by B. Sroysang, Konuralp J. Math. Vol:1, No.2 (2013), 30-32.
• [7] B. Sroysang, More on the Diophantine equation 8x +19y = z2, Int. J. Pure Appl. Math. Vol:81, No.4 (2013), 601-604.
• [8] A. Suvarnamani, A. Singta, S. Chotchaisthit, On two Diophantine Equations 4x +7y = z2 and 4x +11y = z2, Sci. Technol. RMUTT J. Vol:1 (2011), 25-28.