Fixed Points and Cycles of the Kaprekar Transformation: 1. Odd Bases
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We extend Yamagami and Matsui's theory of the Kaprekar transformation in
base 3 to higher odd bases. The structure of primitive proper, primitive
non-proper and general cycles appears in all odd bases, but many other
features differ according to whether the base is congruent to 1 or 3
modulo 4. In the latter case, cycles are derived from subgroups and
cosets in multiplicative groups modulo odd numbers. We examine cycles
and fixed points in bases 5 and 7 in some detail, and make some broad
observations relating to higher bases. There also exist many cycles
outside the primitive/general structure.
Full version: pdf,
(Concerned with sequences
Received April 21 2022; revised versions received July 17 2022; July 18 2022; July 21 2022.
Published in Journal of Integer Sequences,
July 22 2022.
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