Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Lower bounds for the total stopping time of $3x + 1$ iterates


Authors: David Applegate and Jeffrey C. Lagarias
Journal: Math. Comp. 72 (2003), 1035-1049
MSC (2000): Primary 11B83; Secondary 11Y16, 26A18, 37A45
DOI: https://doi.org/10.1090/S0025-5718-02-01425-4
Published electronically: June 6, 2002
MathSciNet review: 1954983
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The total stopping time $\sigma_{\infty}(n)$ of a positive integer $n$ is the minimal number of iterates of the $3x+1$ function needed to reach the value $1$, and is $+\infty$ if no iterate of $n$ reaches $1$. It is shown that there are infinitely many positive integers $n$ having a finite total stopping time $\sigma_{\infty}(n)$ such that $\sigma_{\infty}(n) > 6.14316 \log n.$ The proof involves a search of $3x +1$ trees to depth 60, A heuristic argument suggests that for any constant $\gamma < \gamma_{BP} \approx 41.677647$, a search of all $3x +1$ trees to sufficient depth could produce a proof that there are infinitely many $n$ such that $\sigma_{\infty}(n)>\gamma\log n.$It would require a very large computation to search $3x + 1$ trees to a sufficient depth to produce a proof that the expected behavior of a ``random'' $3x +1$ iterate, which is $\gamma=\frac{2}{\log 4/3} \approx 6.95212,$occurs infinitely often.


References [Enhancements On Off] (What's this?)

  • 1. D. Applegate and J. C. Lagarias, Density bounds for the $3x+1$ problem I. Tree-search method, Math. Comp. 64 (1995), 411-426. MR 95c:11024
  • 2. -, Density bounds for the $3x+1$ problem II. Krasikov inequalities, Math. Comp. 64 (1995), 427-438. MR 95c:11025
  • 3. -, The distribution of $3x+1$ trees, Experimental Math. 4 (1995), 101-117. MR 97e:11033
  • 4. K. Borovkov and D. Pfeifer, Estimates for the Syracuse problem via a probabilistic model, Theory Probab. Appl. 45 (2000), 300-310.
  • 5. R. E. Crandall, On the ``$3x+1$'' problem, Math. Comp. 32 (1978), 1281-1292. MR 58:494
  • 6. J. C. Lagarias, The $3x+1$ problem and its generalizations, Amer. Math. Monthly 92 (1985), 3-23. MR 86i:11043
  • 7. J. C. Lagarias and A. Weiss, The $3x+1$ problem: Two stochastic models, Ann. Applied Prob. 2 (1992), 229-261. MR 92k:60159
  • 8. H. Müller, Das `$3n+1$' Problem, Mitteilungen der Math. Ges. Hamburg 12 (1991), 231-251. MR 93c:11053
  • 9. T. Oliveira e Silva, Maximum excursion and stopping time record-holders for the $3x+1$problem: computational results, Math. Comp. 68, No. 1 (1999), 371-384. MR 2000g:11015
  • 10. D. W. Rawsthorne, Imitation of an iteration, Math. Mag. 58 (1985), 172-176. MR 86i:40001
  • 11. E. Roosendaal, private communication. See also: On the $3x+1$ problem, electronic manuscript, available at http://personal.computrain.nl/eric/wondrous
  • 12. S. Wagon, The Collatz problem, Math. Intelligencer 7 (1985), 72-76. MR 86d:11103
  • 13. G. J. Wirsching, The dynamical system generated by the $3n+1$ function, Lecture Notes in Math. No. 1681, Springer-Verlag: Berlin 1998. MR 99g:11027

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11B83, 11Y16, 26A18, 37A45

Retrieve articles in all journals with MSC (2000): 11B83, 11Y16, 26A18, 37A45


Additional Information

David Applegate
Affiliation: AT&T Laboratories, Florham Park, New Jersey 07932-0971
Email: david@research.att.com

Jeffrey C. Lagarias
Affiliation: AT&T Laboratories, Florham Park, New Jersey 07932-0971
Email: jcl@research.att.com

DOI: https://doi.org/10.1090/S0025-5718-02-01425-4
Received by editor(s): February 6, 2001
Received by editor(s) in revised form: June 7, 2001
Published electronically: June 6, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society