Running modulus recursions

Dearden, Bruce ; Metzger, Jerry

Journal of integer sequences, Tome 13 (2010) no. 1

Résumé

Fix integers $b \ge 2$ and $k \ge 1$. Define the sequence ${z_{n}}$ recursively by taking $z_{0}$ to be any integer, and for $n \ge 1$, taking $z_{n}$ to be the least nonnegative residue of $bz_{n-1}$ modulo $(n+k)$. Since the modulus increases by 1 when stepping from one term to the next, such a definition will be called a running modulus recursion or $rumor$ for short. While the terms of such sequences appear to bounce around irregularly, empirical evidence suggests the terms will eventually be zero. We prove this is so when one additional assumption is made, and we conjecture that this additional condition is always met.

Détail

Classification : 11B37, 11B50
Keywords: recurrence sequence, recurrence relation modulo m, josephus problem, running modulus recursion