On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

2021 ◽  
pp. 1-27
Author(s):  
Harold S. Erazo ◽  
Carlos A. Gómez ◽  
Florian Luca
Keyword(s):  
Pell Equation ◽  
Number Of Solutions ◽  
Pell Equations ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences ◽  
Powers Of 2 ◽  
Integer Solutions

In this paper, we show that if [Formula: see text] is the [Formula: see text]th solution of the Pell equation [Formula: see text] for some non-square [Formula: see text], then given any integer [Formula: see text], the equation [Formula: see text] has at most [Formula: see text] integer solutions [Formula: see text] with [Formula: see text] and [Formula: see text], except for the only pair [Formula: see text]. Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

Estimating the number of solutions of systems of nonlinear equations with linear recurring arguments by the spectral method

2017 ◽  
Vol 27 (4) ◽  
Author(s):  
Oleg V. Kamlovskiy
Keyword(s):  
Spectral Method ◽  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
Galois Rings ◽  
Number Of Solutions ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences

Abstract A general approach is proposed for obtaining estimates of the number of solutions of systems of nonlinear equations. Final estimates are established in the case when the arguments of functions in the system are the signs of linear recurrent sequences over Galois rings.

scholarly journals Primality tests, linear recurrent sequences and the Pell equation

2021 ◽  
Author(s):  
Danilo Bazzanella ◽  
Antonio Di Scala ◽  
Simone Dutto ◽  
Nadir Murru
Keyword(s):  
Pell Equation ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences

scholarly journals On the Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2

2021 ◽  
Vol 27 (2) ◽  
pp. 88-100
Author(s):  
Qiongzhi Tang ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Pell Equation ◽  
Intersection Angle ◽  
Integer Solutions ◽  
Two Sides

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle $120^\circ$ or $60^\circ$.

scholarly journals On the number of solutions to systems of Pell equations

2007 ◽  
Vol 125 (2) ◽  
pp. 356-392 ◽  
Author(s):  
Mihai Cipu ◽  
Maurice Mignotte
Keyword(s):  
Number Of Solutions ◽  
Pell Equations

The Positive Integer Solutions of a Diophantine Equation

2015 ◽  
Vol 713-715 ◽  
pp. 1483-1486
Author(s):  
Yi Wu ◽  
Zheng Ping Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Pell Equation ◽  
Integer Solutions ◽  
Basic Equations

In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

On the $x$-coordinates of Pell equations which are Fibonacci numbers

2018 ◽  
Vol 122 (1) ◽  
pp. 18 ◽  
Author(s):  
Florian Luca ◽  
Alain Togbé
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Pell Equation ◽  
Pell Equations

For an integer $d>2$ which is not a square, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.

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