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Book Details2010; 344 ppMSC: Primary 11; 37; Secondary 68;
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 \(x < 5.4 \cdot 10^{18}\). 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.ReadershipGraduate students and research mathematicians interested in number theory.

Table of Contents

Part I. Overview and introduction

1. The $3x+1$ problem: An overview

2. The $3x+1$ problem and its generalizations

Part II. Survey papers

3. A $3x+1$ Survey: Number theory and dynamical systems

4. Generalized $3x+1$ mappings: Markov chains and ergodic theory

5. Generalized $3x+1$ functions and the theory of computation

Part III. Stochastic modelling and computation papers

6. Stochastic models for the $3x+1$ and $5x+1$ problems and related problems

7. Empirical verification of the $3x+1$ and related conjectures

Part IV. Reprinted early papers

8. Cyclic sequences and Frieze patterns (The Fourth Felix Behrend Memorial Lecture)

9. Unpredictable iterations

10. Iteration of the numbertheoretic function $f(2n)=n,f(2n+1)=3n+2$

11. Don’t try to solve these problems!

12. On the motivation and origin of the $(3n+1)$problem

13. FRACTRAN: A simple universal programming language for arithmetic

Part V. Annotated bibliography

14. The $3x+1$ problem: An annotated bibliography (1963–1999)


Additional Material

Reviews

Let me cut to the chase: Lagarias has assembled a fantastic book on a fascinating topic, and it is the type of book that the mathematical community could use more of. The book assembles a variety of articles written about the topic over the last forty years, coming to the material from different directions and using different flavors of mathematics, all in service of trying to solve this problem.
MAA Reviews 
[This book] contains...two surveys by editor Lagarias...the world's foremost expert. [It also contains] a tremendously useful, richly annotated bibliography...[to] round out the volume. ....A must for all libraries. Highly recommended.
D.V. Feldman, Choice 
[T]his book is a thorough account of an open and challenging problem.
Vincente Muqoz, The European Mathematical Society


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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 \(x < 5.4 \cdot 10^{18}\). 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.
Graduate students and research mathematicians interested in number theory.

Part I. Overview and introduction

1. The $3x+1$ problem: An overview

2. The $3x+1$ problem and its generalizations

Part II. Survey papers

3. A $3x+1$ Survey: Number theory and dynamical systems

4. Generalized $3x+1$ mappings: Markov chains and ergodic theory

5. Generalized $3x+1$ functions and the theory of computation

Part III. Stochastic modelling and computation papers

6. Stochastic models for the $3x+1$ and $5x+1$ problems and related problems

7. Empirical verification of the $3x+1$ and related conjectures

Part IV. Reprinted early papers

8. Cyclic sequences and Frieze patterns (The Fourth Felix Behrend Memorial Lecture)

9. Unpredictable iterations

10. Iteration of the numbertheoretic function $f(2n)=n,f(2n+1)=3n+2$

11. Don’t try to solve these problems!

12. On the motivation and origin of the $(3n+1)$problem

13. FRACTRAN: A simple universal programming language for arithmetic

Part V. Annotated bibliography

14. The $3x+1$ problem: An annotated bibliography (1963–1999)

Let me cut to the chase: Lagarias has assembled a fantastic book on a fascinating topic, and it is the type of book that the mathematical community could use more of. The book assembles a variety of articles written about the topic over the last forty years, coming to the material from different directions and using different flavors of mathematics, all in service of trying to solve this problem.
MAA Reviews 
[This book] contains...two surveys by editor Lagarias...the world's foremost expert. [It also contains] a tremendously useful, richly annotated bibliography...[to] round out the volume. ....A must for all libraries. Highly recommended.
D.V. Feldman, Choice 
[T]his book is a thorough account of an open and challenging problem.
Vincente Muqoz, The European Mathematical Society