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

Schedler, Zak

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

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Brock_Schedler_Zak_2019.pdf (408.93 KB)

Authors

Schedler, Zak

Publisher

Brock University

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 $y 2= p x (A x 2− C), C ∈ 2,±1,±4 . W h e r e p i s a n o d d p r i m e a n d A i s a n i n t e g e r g r e a t e r t h a n 1. T h e c a s e w h e r e C =−2 i s a l r e a d y c o m p l e t e, w h i c h w e g o o v e r i n d e t a i l h e r e . W e l o o k t h r o u g h e x a m p l e s o f D i o p h a n t i n e e q u a t i o n s s t a r t i n g w i t h l i n e a r D i o p h a n t i n e e q u a t i o n s . W e t h e n l o o k a t P e l l ′ s e q u a t i o n,$ 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 $y 2= p x (A x 2− C)$ . 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}.