The Ultimate Challenge: The $3x+1$ ProblemJeffrey C. Lagarias American Mathematical Society, 19 d’abr. 2023 - 344 pàgines 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. |
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2-adic integers 3x+1 conjecture 3x+1 function 3x+1 map 3x+1 problem Acta Arithmetica arithmetic behavior Benford branching random walk cellular automata classes mod Collatz function Collatz mapping Collatz problem congruence classes constant counts the number cycles d-adic defined denote distribution divergent trajectories encode equation ergodic theory example exists Felix Behrend finite forward iteration FRACTRAN program frieze patterns given graph H. S. M. Coxeter halting problem infinite initial inverse iterates J. C. Lagarias J. H. Conway Klarner Lagarias Lagarias and Weiss log2 Lothar Collatz Markov chain Math Mathematics maximum excursion Minsky natural density node Note number of iterates Number Theory odd integers paper studies periodic orbits permutation positive integers prediction proof proved random walk random walk model record-holders recursive relatively prime type residue classes set of integers shows stochastic models tag system Theorem total stopping tree Turing machines Ulam undecidable universal vector Wirsching