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2002 •
2005 •
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.
2008 •
2008 •
Lecture Notes in Mathematics
Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures2008 •
Conformal Geometry and Dynamics of the American Mathematical Society
Nearly Euclidean Thurston maps2012 •
Mathematics of Computation
New techniques for bounds on the total number of prime factors of an odd perfect number2007 •
Journal Algebra Discrete Math.
Weak comultiplication modules over a pullback of commutative local Dedekind domains2009 •
p-q-adic combinatorial function theory
Expansion of the p-adic to the p-q-adic combinatorial function theory2017 •