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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: [email protected]
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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

References

1. Akin, E., Why is the 3x + 1 problem so hard? Math. Magazine, to appear.Google Scholar
2. Bernstein, D. J., A non-iterative 2-adic statement of the 3x+1 conjecture, Proc. Amer. Math. Soc. 121(1994), 405–408.Google Scholar
3. Boyle, M., Franks, J. and Kitchens, B., Automorphisms of one-sided subs hifts of finite type, Ergod. Th. Dyn. Sys. 10(1990), 421–449.Google Scholar
4. Crandall, R. E., On the ‘3x + 1’ problem, Math. Comp. 32(1978), 1281–1292.Google Scholar
5. Franco, Z. and Pomerance, C., On a conjecture of Crandall concerning the QX + 1 problem, Math. Comp. 49(1995), to appear.Google Scholar
6. Hedlund, G., Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3(1969), 320–375.Google Scholar
7. Heppner, E., Eine Bemerkungzum Hasse-Syracuse Algorithmus, Arch. Math. 31(1978), 317—320.Google Scholar
8. Lagarias, J. C., The 3x + 1 problem and its generalizations, Amer. Math. Monthly 92(1985), 3—23.Google Scholar
9. Lagarias, J. C., The set of rational cycles for the 3x + 1 problem, Acta Arithmetica 56(1990), 33–53.Google Scholar
10. Müller, H., Das ‘3n + 1’ Problem, Mitteilungen der Math. Ges. Hamburg 12(1991), 231–251.Google Scholar
11. Müller, H., Über eine Klasse 2-adischer Funktionen im Zussamenhang mit dem “3x+1“ -Problem, Abh. Math. Sem. Univ. Hamburg 64(1994), 293–302.Google Scholar
12. Steiner, R. P., On the “QX+1 problem, “ Q odd, I, II, Fibonacci Quart. 19(1981), 285–288, 293–296 Google Scholar