科拉兹主题变奏曲
Variations on a Theme of Collatz
摘要：
考虑根据以下简单规则从正整数$n_{\ell }$生成新的正整数$n_{\ell +1}$的递归关系：如果整数$n_{\ell }$是奇数，则为$n_{\ell +1}=3n_{\ell }+1$ ;如果整数$n_{\ell }$是偶数，则为$n_{\ell +1}=n_{\ell }/2$。所谓的Collatz猜想指出，从任何正整数$N$开始，由这些规则的持续应用所表征的递归最终在循环$4,$ $2,1$中结束。这个猜想通常被认为是正确的（基于广泛的数值检验），但它尚未被证明。在本文中--基于Collatz猜想确实成立的假设--我们提出了它的一个相当简单的扩展，它包含了将所有自然数分成$3$不相交类的可能性，我们对每个类进行猜想--基于（不是很广泛）数字检查-- $1/3$的所有自然数属于;或者，在某种程度上等价于$2$不相交类，我们猜想所有自然数的$1/3$和$2/3$分别属于这些不相交类。
英文摘要：
Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer $n_{\ell }$ is even, $n_{\ell +1}=n_{\ell }/2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper -- based on the assumption that the Collatz conjecture is indeed true -- we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture -- on the basis of (not very extensive) numerical checks -- $1/3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1/3$ and $2/3$ of all natural numbers belong.
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