Abstract:

For Lucas sequences of the first kind $(u_n)_{n\ge 0}$ and second kind $(v_n)_{n\ge 0}$ defined as usual by $u_n=(\alpha ^n-\beta ^n)/(\alpha -\beta )$, $v_n=\alpha ^n+\beta ^n$, where $\alpha$ and $\beta$ are either integers or conjugate quadratic integers, we describe the sets $\{n\in\mathbb{N}:n$    divides $u_n\}$ and $\{n\in\mathbb{N}:n$    divides $v_n\}$. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called basic number, which can only be $1$, $6$ or $12$, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.