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The 3x + 1 Conjugacy Map

Published online by Cambridge University Press:  20 November 2018

Daniel J. Bernstein  and
Jeffrey C. Lagarias
Daniel J. Bernstein
Affiliation:
Department of Mathematics, University of California, Berkeley, CA, USA 94720
Jeffrey C. Lagarias
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ, USA 07974 e-mail: jcl@research.att.com
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Abstract

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The 3x+1 map T and the shift map S are defined by T(x) = (3x + 1)/2 for x odd, T(x) = x/2 for x even, while S(x) = (x − 1)/2 for x odd, S(x) = x/2 for x even. The 3x + 1 conjugacy map Φ on the 2-adic integers Z2 conjugates S to T, i.e., Φ o S o Φ-1 = T. The map Φ mod 2n induces a permutation Φn on Z/2nZ. We study the cycle structure of Φn. In particular we show that it has order 2n − 4 for n ≥ 6. We also count 1-cycles of Φn for n up to 1000; the results suggest that Φ has exactly two odd fixed points. The results generalize to the ax + b map, where ab is odd.

Keywords

11B75
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 6 , 01 December 1996 , pp. 1154 - 1169
Copyright
Copyright © Canadian Mathematical Society 1996

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