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 查看全文>>