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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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