Diophantine Equation with Arithmetic functions and Binary recurrent sequences

TL;DR

This thesis investigates Diophantine equations involving binary recurrent sequences and arithmetic functions, establishing new bounds, analyzing intersections, and proving the non-existence of Lehmer numbers in certain sequences.

Contribution

It provides new results on Diophantine equations with recurrence sequences and arithmetic functions, including bounds on intersections and proofs related to Lehmer's conjecture.

Findings

Effective bounds on sequence intersections

No Lehmer numbers in Lucas and Pell sequences

Solutions involving Fibonacci, Lucas, and Pell sequences

Abstract

This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $\varphi(|au_n |)=|bv_m|,$ where $\varphi$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ are two non-degenerate binary recurrence sequences and $a,b$ some positive integers. More precisely, we study problems involving members of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler's function remain in the same sequence. We particularly study the case when $\{u_n\}_{n\geq 0}$ is the Fibonacci sequence $\{F_n\}_{n\geq 0}$, the Lucas sequences $\{L_n\}_{n\geq 0}$ or the Pell sequence $\{P_n\}_{n\geq 0}$ and its companion $\{Q_n\}_{n\geq 0}$. Secondly, we look of Lehmer's conjecture on some recurrence sequences. Recall that a composite number $N$ is said to be Lehmer if $\varphi(N)\mid N-1$. We prove that there is no Lehmer number neither in the Lucas sequence $\{L_n\}_{n\geq 0}$ nor in the Pell sequence $\{P_n\}_{n\geq 0}$. The main tools used in this thesis are lower bounds for linear forms in logarithms of algebraic numbers, the so-called Baker-Davenport reduction method, continued fractions, elementary estimates from the theory of prime numbers and sieve methods.