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Lower bounds for the total stopping time of $$3x + 1$$ iterates. (English) Zbl 1052.11017
Summary: 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 $$\gamma < \gamma_{BP} \approx 41.677647$$, 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.

##### MSC:
 11B83 Special sequences and polynomials 11Y16 Number-theoretic algorithms; complexity 26A18 Iteration of real functions in one variable 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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##### References:
  David Applegate and Jeffrey C. Lagarias, Density bounds for the 3\?+1 problem. I. Tree-search method, Math. Comp. 64 (1995), no. 209, 411 – 426. · Zbl 0820.11006  David Applegate and Jeffrey C. Lagarias, Density bounds for the 3\?+1 problem. II. Krasikov inequalities, Math. Comp. 64 (1995), no. 209, 427 – 438. · Zbl 0820.11006  David Applegate and Jeffrey C. Lagarias, The distribution of 3\?+1 trees, Experiment. Math. 4 (1995), no. 3, 193 – 209. · Zbl 0868.11012  K. Borovkov and D. Pfeifer, Estimates for the Syracuse problem via a probabilistic model, Theory Probab. Appl. 45 (2000), 300-310. · Zbl 0984.60050  R. E. Crandall, On the ”3\?+1” problem, Math. Comp. 32 (1978), no. 144, 1281 – 1292. · Zbl 0395.10013  Jeffrey C. Lagarias, The 3\?+1 problem and its generalizations, Amer. Math. Monthly 92 (1985), no. 1, 3 – 23. · Zbl 0566.10007 · doi:10.2307/2322189 · doi.org  J. C. Lagarias and A. Weiss, The 3\?+1 problem: two stochastic models, Ann. Appl. Probab. 2 (1992), no. 1, 229 – 261. · Zbl 0742.60027  Helmut Müller, Das ”3\?+1”-Problem, Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 231 – 251 (German). Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. · Zbl 0773.11018  Tomás Oliveira e Silva, Maximum excursion and stopping time record-holders for the 3\?+1 problem: computational results, Math. Comp. 68 (1999), no. 225, 371 – 384. · Zbl 0919.11020 · doi:10.1090/S0025-5718-99-01031-5 · doi.org  Daniel A. Rawsthorne, Imitation of an iteration, Math. Mag. 58 (1985), no. 3, 172 – 176. · Zbl 0587.10030 · doi:10.2307/2689917 · doi.org  E. Roosendaal, private communication. See also: On the $$3x+1$$ problem, electronic manuscript, available at http://personal.computrain.nl/eric/wondrous  Stan Wagon, The Collatz problem, Math. Intelligencer 7 (1985), no. 1, 72 – 76. · Zbl 0566.10008 · doi:10.1007/BF03023011 · doi.org  Günther J. Wirsching, The dynamical system generated by the 3\?+1 function, Lecture Notes in Mathematics, vol. 1681, Springer-Verlag, Berlin, 1998. · Zbl 0892.11002
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