The Dynamical System Generated by the 3n+1 Function
The 3n+1 function T is defined by T(n)=n/2 for n even, and T(n)=(3n+1)/2 for n odd. The famous 3n+1 conjecture, which remains open, states that, for any starting number n>0, iterated application of T to n eventually produces 1. After a survey of theore
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Gunther J. Wirsching
The Dynamical System Generated by the 3n+l Function
Springer
Author Gunther J. Wirsching Katholische Universitat Eichstatr Mathematisch-Geographische Fakultat D-85071 Eichstatt, Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Wirsching, Gunther: The dynamical system generated by the 30 + 1 function / Gunther J. Wirsching. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo: Springer, 1998 (Lecture notes in mathematics; 1681) ISBN 3-540-63970-5
Mathematics Subject Classification (1991): 11B37, 60C05, 60B 10, 11K41 ISSN 0075-8434 ISBN 3-540-63970-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10649733 46/3143-543210 - Printed on acid-free paper
THE DYNAMICAL SYSTEM ON THE NATURAL NUMBERS GENERATED BY THE 3n
+1
FUNCTION
Table of Contents Introduction Chapter I. Some ideas around 3n + 1 iterations 1. The problem 2. About the origin of the problem 3. Empirical investigations and stochastic models 4. Related functions and generalizations 5. Some formulae describing the iteration 6. Numbers with finite stopping time 7. Asymptotics of predecessor sets 8. Consecutive numbers with the same height 9. Cycles 10. Binary sequences and 2-adic analysis 11. Reduction to residue classes and other sets 12. Formal languages 13. Functional equations 14. A continuous extension to the real line Chapter II. Analysis of the Collatz graph 1. Directed graphs and dynamical systems on l'J Directed graphs The Collatz graph The size of a subset of l'J 2. Encoding of predecessors by admissible vectors Encoding a path in the Collatz graph Concatenation of integer vectors Tracing back integer vectors in the rationals Admissible integer vectors 3. Some properties of admissible vectors Recognizing admissible vectors Extending admissible vectors Similar integer vectors
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