On Some Problems Concerning Integer Recurring Sequences

This thesis focuses on the arithmetic of certain recurring sequences. We consider the sequences $a^n-1$, the Fibonacci sequence, and more general Lucas sequence. \\The first section is inspired by Artin's primitive root conjecture. We prove that if $a,b$ are multiplicatively independent, then for almost all prime numbers $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{\frac{8}{15}+ \epsilon(p)}$. We also show that for infinitely many primes $p$, the order of the Fibonacci Sequence is as large as possible. \\ In the second section we prove the existence and continuity of the distribution functions of the density of normal and primitive elements in a finite field and the reciprocal sum of divisors of Lucas sequences.