The Ultimate Challenge

The $3x+1$ Problem

Buch | Softcover

344 Seiten

2010
American Mathematical Society (Verlag)
978-1-4704-7289-4 (ISBN)

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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.

Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI.

Overview and introduction: J. Lagarias, The $3x+1$ problem: An overview
J. Lagarias, The $3x+1$ problem and its generalizations
Survey papers: M. Chamberland, A $3x+1$ Survey: Number theory and dynamical systems
K. R. Matthews, Generalized $3x+1$ mappings: Markov chains and ergodic theory
P. Michel and M. Margenstern, Generalized $3x+1$ functions and the theory of computation
Stochastic modelling and computation papers: A. V. Kontorovich and J. Lagarias, Stochastic models for the $3x+1$ and $5x+1$ problems and related problems
T. O. e Silva, Empirical verification of the $3x+1$ and related conjectures
Reprinted early papers: H. S. M. Coxeter, Cyclic sequences and Frieze patterns (The Fourth Felix Behrend Memorial Lecture)
J. H. Conway, Unpredictable iterations
C. J. Everett, Iteration of the number-theoretic function $f(2n)=n,f(2n+1)=3n+2$
R. K. Guy, Don't try to solve these problems!
L. Collatz, On the motivation and origin of the $(3n+1)$-problem
J. H. Conway, FRACTRAN: A simple universal programming language for arithmetic
Annotated bibliography: J. Lagarias, The $3x+1$ problem: An annotated bibliography (1963-1999)

Erscheinungsdatum	10.03.2023
Reihe/Serie	Miscellaneous Book Series
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
ISBN-10	1-4704-7289-9 / 1470472899
ISBN-13	978-1-4704-7289-4 / 9781470472894
Zustand	Neuware