Embedding the \(3x+1\) conjecture in a \(3x+d\) context. (English) Zbl 0918.11008

Belaga, Edward G.; Mignotte, Maurice

Exp. Math. 7, No. 2, 145-151 (1998).

The \(3x+1\) problem concerns the behavior of iterations of the function \(T:{\mathbb{N}}^*\to{\mathbb{N}}^*\) (where \({\mathbb{N}}^*=\{1,2,\ldots\}\)), \(T(n)=n/2\) if \(n\) even, \(T(n)=(3n+1)/2\) if \(n\) odd. Now let \(d\) be an odd integer, and denote by \(S_d(n)\) the highest odd factor of \(3n+d\). Then, for a starting number \(x\), the sequence of iterates \((S_1(x),S_1^2(x),\ldots)\) consists of the odd elements of the \(3n+1\) trajectory \((x,T(x),T^2(x),\ldots)\). Now concentrate on \(S_d\). For an integer \(k\geqslant 1\), a set \(\{n,S_d(n),S_d^2(n),\ldots,S_d^{k-1}(n)\}\) is called a \(k\)-cycle if \(S_d^k(n)=n\) and \(S_d^j(n)\neq n\) for \(0<j<k\). Denote by \({\mathbb{D}}\) the set of odd positive integers which are not divisible by \(3\), then the following proposition is proved:
For any positive integer \(k\) there exist \(d,n\in\mathbb{D}\) such that \(n\) belongs to a cycle of length \(k\) for \(S_d\). Using a deep result of A. Baker and G. Wüstholz [ J. Reine Angew. Math. 442, 19-62 (1993; Zbl 0788.11026)], the authors prove: For given \(d\in\mathbb{D}\cup\{-1\}\) and a given positive integer \(k\), the number of \(k\)-periodic points of \(S_d\) is finite. In the last section, a two-to-one map \(W:\mathbb{D}\to\mathbb{D}\) is construced which can be considered as a ‘normalized’ version of \(S_1\).

Reviewer: G.Wirsching (Eichstätt)

MSC:

11B37

Recurrences

Keywords:

Collatz problem; 3x+1 problem; generalizations of the 3x+1 problem; finite cycles conjecture; bounds for linear forms in logarithms

Citations:

Zbl 0788.11026

References:

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