Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.6

Running Modulus Recursions

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


Fix integers b ≥ 2 and k ≥ 1. Define the sequence {zn} recursively by taking z0 to be any integer, and for n ≥ 1, taking zn to be the least nonnegative residue of bzn-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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