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

                 bruce.dearden@und.edu
                  joel.iiams@und.edu
                 jerry.metzger@und.edu

Abstract

For an integer b ≥ 2 and for x ∈ [0, 1), define ρ_{b}(x) = ∑_{n=0}^{∞} \frac{\{\{b^{n}x\}\}}{b^{n}}, where \{\{t\}\} denotes the fractional part of the real number t. A number of properties of ρ_{b} are derived, and then a connection between ρ_{b} and the rumor conjecture is established. To form a rumor sequence {z_{n}}, first select integers b ≥ 2 and k ≥ 1. Then select an integer z_{0}, and for n ≥ 1 let z_{n} = bz_{n−1} mod (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 ρ_{b} is shown to be equivalent to the rumor conjecture.