Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba¹

¹ Department of Mathematics, Computer Science and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 98 (2003) no. 2, pp. 163-169

Résumé

Consider a recurrence sequence $(x_k)_{k\in{\mathbb Z}}$ of integers satisfying $ x_{k+n}=a_{n-1}x_{k+n-1}+\ldots +a_1x_{k+1}+a_0x_k $, where $a_0,a _1,\ldots,a_{n-1}\in{\mathbb Z}$ are fixed and $a_0\in\{-1,1\}$. Assume that $x_k>0$ for all sufficiently large $k$. If there exists $k_0\in{\mathbb Z}$ such that $ x_{k_0}0 $ then for each negative integer $-D$ there exist infinitely many rational primes $q$ such that $q\,|\, x_k$ for some $k\in{\mathbb N}$ and $(\frac{-D}{q})=-1$.

Détail

DOI : 10.4064/cm98-2-3

Keywords: consider recurrence sequence mathbb integers satisfying n n ldots where ldots n mathbb fixed assume sufficiently large there exists mathbb each negative integer d there exist infinitely many rational primes mathbb frac d