=4in

A Function Related to the
Rumor Sequence Conjecture Bruce Dearden, Joel Iiams, and Jerry Metzger
University of North Dakota
Department of Mathematics
Witmer Hall Room 313
101 Cornell Street Stop 8376
Grand Forks, ND 58202-8376
USA
mailto:bruce.dearden@und.edubruce.dearden@und.edu
mailto:joel.iiams@und.edujoel.iiams@und.edu
mailto:jerry.metzger@und.edujerry.metzger@und.edu

Abstract:

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.