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A density estimate for the \( 3x+1\) problem. (English) Zbl 0797.11027

The “\(3x+1\)”-problem (or “Collatz”- or “Hasse”- or “Syracuse”- or “Kakutani”-problem) is to prove that for every \(n\in\mathbb{N}\) there exists a \(k\) with \(T^{(k)}(n)=1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\). This hypothesis is equivalent to the statement that for every positive integer \(y\) there is \(n\) such that \(T^{(n)}<y\). C. J. Everett in [Adv. Math. 25, 42-45 (1977; Zbl 0352.10001)] and R. Terras in [Acta Arith. 30, 241-252 (1976; Zbl 0348.10037)] proved that the asymptotic density of \(\{y \in \mathbb{N} \mid \exists n\) with \(T^{(n)}<y\}\) equals 1.
In this paper the author shows that the inequality could be replaced by a stronger one, namely he proves that the asymptotic density of \(\{y \in \mathbb{N} \mid \exists n\) with \(T^{(n)}<y^ c\}\) equals 1 too, where \(c\) is any real number greater than \(\log_ 4 3(=0.79248125 \dots)\).

MSC:

11B83 Special sequences and polynomials
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References:

[1] ALLOUCHE J.-P.: Sur la conjecture de ’Syracuse-Kakutani-Collatz’. Seminar de Théorie des Nombres. Exp. No. 9, CNRS Talence, 1978-1979.
[2] DOLAN J. M., GILMAN A. F., MANICKAM S.: A generalization of Everett’s result on the Collatz 3x + 1 problem. Adv. in Appl. Math. 8 (1987), 405-409. · Zbl 0648.10009 · doi:10.1016/0196-8858(87)90018-2
[3] EVERETT C. J.: Iteration of the number theoretic function f(2n)=n, f(2n + 1)= 3n + 2. Adv. Math. 25 (1977), 42-45. · Zbl 0352.10001 · doi:10.1016/0001-8708(77)90087-1
[4] HEPPNER E.: Eine Bemerkung zum Hasse-Syracuse-Algorithmus. Arch. Math. (Basel) 31 (1978), 317-320. · Zbl 0377.10027 · doi:10.1007/BF01226454
[5] LAGARIAS J. C.: The 3x + 1 problem and its generalizations. Amer. Math. Monthly 92 (1985), 3-23. · Zbl 0566.10007 · doi:10.2307/2322189
[6] TERRAS R.: A stopping time on the positive integers. Acta Arith. XXX (1976), 241-252. · Zbl 0348.10037
[7] TERRAS R.: On the existence of a density. Acta Arith. XXXV (1979), 101-102. · Zbl 0413.10052
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