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Running Modulus Recursions
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Bruce Dearden 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

**Abstract:**

Fix integers *b* ≥ 2 and *k* ≥ 1.
Define the sequence {*z*_{n}}
recursively by taking *z*_{0} to be any integer,
and for *n* ≥ 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.

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Received December 2 2009;
revised version received January 12 2010.
Published in *Journal of Integer Sequences*, January 14 2010.

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