The terms in Lucas sequences divisible by their indices

Smyth, Chris

Journal of integer sequences, Tome 13 (2010) no. 2.

Résumé

Summary: 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.

Détail

Classification : 11B39
Keywords: Lucas sequences, indices