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February, 1992 The 3x+1 Problem: Two Stochastic Models
J. C. Lagarias, A. Weiss
Ann. Appl. Probab. 2(1): 229-261 (February, 1992). DOI: 10.1214/aoap/1177005779

Abstract

The 3x+1 problem concerns the behavior under iteration of the function T:Z+Z+ defined by T(n)=n/2 if n is even and T(n)=(3n+1)/2 if n is odd. The 3x+1 conjecture asserts that for each n1 some k exists with T(k)(n)=1; let σ(n) equal the minimal such k if one exists and + otherwise. The behavior of σ(n) is irregular and seems to defy simple description. This paper describes two kinds of stochastic models that mimic some of its features. The first is a random walk that imitates the behavior of T(mod2j); the second is a family of branching random walks that imitate the behavior of T1(mod3j). For these models we prove analogues of the conjecture that limsupn(σ(n)/log(n))=γ for a finite constant γ. Both models produce the same constant γ041.677647. Predictions of the stochastic models agree with empirical data for the 3x+1 problem up to 1011. The paper also studies how many n have σ(n)=k as k and estimates how fast t(n)=max(T(k)(n):k0) grows as n.

Citation

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J. C. Lagarias. A. Weiss. "The 3x+1 Problem: Two Stochastic Models." Ann. Appl. Probab. 2 (1) 229 - 261, February, 1992. https://doi.org/10.1214/aoap/1177005779

Information

Published: February, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0742.60027
MathSciNet: MR1143401
Digital Object Identifier: 10.1214/aoap/1177005779

Subjects:
Primary: 11A99
Secondary: 26A18 , 60F10 , 60J85

Keywords: , Branching random walk , large deviations

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 1 • February, 1992
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