Non-existence of Solutions of Diophantine Equations of the Form &lt;img src=image/13490640_01.gif&gt;

Renz Jimwel S Mina,Jerico B Bacani

doi:10.13189/ms.2019.070304

Abstract

Numerous researches have been devoted in finding the solutions , in the set of non-negative integers, of Diophantine equations of type (1), where the values p and q are fixed. In this paper, we also deal with a more generalized form, that is, equations of type (2), where n is a positive integer. We will present results that will guarantee the non-existence of solutions of such Diophantine equations in the set of positive integers. We will use the concepts of the Legendre symbol and Jacobi symbol, which were also used in the study of other types of Diophantine equations. Here, we assume that one of the exponents is odd. With these results, the problem of solving Diophantine equations of this type will become relatively easier as compared to the previous works of several authors. Moreover, we can extend the results by considering the Diophantine equations (3) in the set of positive integers.

Highlights

For the past decade, Diophantine equations of type ppxx + qqyy = zz2 have been widely studied for various values of pp and qq
The goal of this paper is to present an easier way of showing that certain Diophantine equations of type ppxx + qqyy = zz2, where pp and qq are fixed positive integers, may fail to have solutions in the set N of positive integers
This is done by using the concepts of Legendre symbol and the Jacobi symbol

Summary

Introduction

In 2011, Suvarnami, et al [2] showed that the Diophantine equation 4xx + 7yy = zz and 4xx + 11yy = zz have no solutions in the N0. The goal of this paper is to present an easier way of showing that certain Diophantine equations of type ppxx + qqyy = zz, where pp and qq are fixed positive integers, may fail to have solutions in the set N of positive integers. This is done by using the concepts of Legendre symbol and the Jacobi symbol

Preliminaries

Main Results

Summary