The Ultimate Challenge: The 3x+1 ProblemJeffrey C. Lagarias American Mathematical Soc., 2010 - 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 |
Continguts
an overview | 3 |
The 3x + 1 problem and its generalizations | 31 |
Number theory and dynamical systems | 57 |
Markov chains and ergodic theory | 79 |
Generalized 3x + 1 functions and the theory of computation | 105 |
Stochastic Modelling and Computation Papers | 131 |
Empirical verification of the 3x + 1 and related conjectures | 189 |
Cyclic sequences and frieze patterns The Fourth Felix Behrend Memorial | 211 |
Unpredictable iterations | 219 |
f2n n f2n +13n + 2 | 225 |
Dont try to solve these problems | 231 |
Lothar Collatz | 241 |
An annotated bibliography 19631999 | 267 |
Altres edicions - Mostra-ho tot
Frases i termes més freqüents
3x+1 problem Acta algorithm appears base behavior bound branching called Collatz function computation concerning condition conjecture consider constant contains continued Conway count cycles defined definition denote density depth described determined distribution divergent trajectories dynamics equation ergodic eventually exactly example excursion exists extension Figure finite fixed given gives graph holds implies infinite initial interesting iterates known Lagarias length limiting Markov Math Mathematics maximum measure natural node Note Number Theory observed obtained orbits particular periodic positive integers prediction present prime probability proof proved question random walk rational reach relatively satisfies sequence shows similar starting statistics steps stochastic models structure symbols Table Theorem theory total stopping trajectories tree Turing machines undecidable universal variable vector volume