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

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 $$S_1$$.

### MSC:

 11B37 Recurrences

Zbl 0788.11026
Full Text:

### References:

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