The Ultimate Challenge: The 3x+1 Problem

by Jeffrey C. Lagarias (editor)

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.

Genres: Mathematics

344 pages, Hardcover

First published January 14, 2011

Reviews

[Nick Black]

http://bit-player.org/2011/dont-try-t... ahhhh the Collatz problem, you dirty old motherfucker. i wasted about three weeks of my life on this gnarly whore back in 2002, and made absolutely no progress, aieeeeee...

[Stuart · Rating 4 out of 5]

The "3x+1 problem," also known as the "Collatz conjecture," can be stated with high-school algebra, and it is believed to be true, but after more than 80 years no mathematician has been able to actually prove that it is true. If you become seduced into pursuing this problem, as I have been, this comprehensive book is the best place to begin serious inquiry. This 2010 collection of essays and articles covers the history (interesting), some simple discussions and examples (where I belong), and some very sophisticated elaborations (beyond me).

[Sylvain Bérubé · Rating 5 out of 5]

Survol complet du problème mathématique non résolu le plus fascinant du monde. Si le problème en question est compréhensible par un enfant de sept ans, des notions mathématiques avancées sont requises pour apprécier ce livre.