A function related to the rumor sequence conjecture

Dearden, Bruce ; Iiams, Joel ; Metzger, Jerry

Journal of integer sequences, Tome 14 (2011) no. 2

Résumé

For an integer $ b\geq 2$ and for $ x\in [0,1)$, define $ \rho_b(x) = \sum_{n=0}^{\infty} \frac{\{\mskip -5mu\{b^nx\}\mskip -5mu\}}{b^n}$, where $ \{\mskip -5mu\{t\}\mskip -5mu\}$ denotes the fractional part of the real number $ t$. A number of properties of $ \rho_b$ are derived, and then a connection between $ \rho_b$ and the rumor conjecture is established. To form a rumor sequence $ \{z_n\}$, first select integers $ b\geq 2$ and $ k\geq 1$. Then select an integer $ z_0$, and for $ n\geq 1$ let $ z_n = bz_{n-1} \bmod{(n+k)}$, where the right side is the least non-negative residue of $ bz_{n-1}$ modulo $ n+k$. The rumor sequence conjecture asserts that all such rumor sequences are eventually 0. A condition on $ \rho_b$ is shown to be equivalent to the rumor conjecture.

Détail

Keywords: recurrence sequence, recurrence relation modulo m, running modulus recursion, takagi function (Related to sequences and )