Upper Bounds for the Number of solutions for the Diophantine Equation $y^2=px(Ax^2-C), C \in {2, \pm 1, \pm 4}$

Abstract

A Diophantine equation is an equation of more than one variable where we are looking for strictly integer solutions. The purpose of this paper is to give a new upper bounds for the number of positive solutions for the Diophantine equation $y^2 = px(Ax^2 − C), C \in {2, ±1, ±4}. Where p is an odd prime and A is an integer greater than 1. The case where C = −2 is already complete, which we go over in detail here. We look through examples of Diophantine equations starting with linear Diophantine equations. We then look at Pell’s equation, $x^2 − Dy^2 = C$ where D and C are natural numbers. We show the continued fraction algorithm and how to use it to solve Pell’s equation. We will look at proofs and lemmas surrounding particular cases of the Diophantine equation $y^2 = px(Ax^2 − C)$. Then focus on finding the upper bounds of the equation. Then we conclude by showing the new upper bounds of the Diophantine equation $y^2 = px(Ax^2 − C), C \in {2, ±1, ±4}.