On the Glide of 3x+1 Problem

Yuyin Yu; Dingyi Pei

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arXiv · 1710.01525

On the Glide of 3x+1 Problem

Yuyin Yu·Dingyi Pei

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Abstract

For any positive integer $n$, define an iterated function $$ f(n)=\left\{\begin{array}{ll} n/2, & \mbox{$n$ even,} \\ 3n+1, & \mbox{$n$ odd.} \end{array} \right. $$ Suppose $k$ (if it exists) is the lowest number such that $f^{k}(n)<n$, and there are $O(n)$ "multiply by three and add one" and $E(n)$ "divide by two" from $n$ to $f^{k}(n)$, then there must be $$ 2^{E(n)-1}<3^{O(n)}<2^{E(n)}. $$ Our results confirm the conjecture proposed by Terras in 1976.

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Yuyin Yu, Dingyi Pei. 2017-10-04. On the Glide of 3x+1 Problem. https://arxiv.org/abs/1710.01525

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