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The ultimate challenge. The $3x+1$ problem. (English) Zbl 1253.11003
Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4940-8/hbk). xiv, 344 p. (2010).

Publisher’s description: The $3x+1$ problem, or Collatz problem, concerns the following seemingly innocent arithmetic procedure applied to integers: If an integer $x$ is odd then “multiply by three and add one”, while if it is even then “divide by two”. The $3x+1$ problem asks whether, starting from any positive integer, repeating this procedure over and over will eventually reach the number 1. Despite its simple appearance, this problem is unsolved. Generalizations of the problem are known to be undecidable, and the problem itself is believed to be extraordinarily difficult.

This book reports on what is known on this problem. It consists of a collection of papers, which can be read independently of each other. The book begins with two introductory papers, one giving an overview and current status, and the second giving history and basic results on the problem. These are followed by three survey papers on the problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computation. The next paper presents results on probabilistic models for behavior of the iteration. This is followed by a paper giving the latest computational results on the problem, which verify its truth for $x<5·4·{10}^{18}$. The book also reprints six early papers on the problem and related questions, by L. Collatz, J. H. Conway, H. S. M. Coxeter, C. J. Everett, and R. K. Guy, each with editorial commentary. The book concludes with an annotated bibliography of work on the problem up to the year 2000.

For more details see the author’s preface http://www.ams.org/bookstore/pspdf/mbk-78-pref.pdf.

The articles have been reviewed independently.

##### MSC:
 11-06 Proceedings of conferences (number theory) 00B15 Collections of articles of miscellaneous specific interest 11-03 Historical (number theory) 11B83 Special sequences of integers and polynomials 37A45 Relations of ergodic theory with number theory and harmonic analysis 01A60 Mathematics in the 20th century