论文标题
关于Collatz一般问题$ QN+1 $
On the Collatz general problem $qn+1$
论文作者
论文摘要
在这项工作中，研究了广义的Collatz问题$ qn+1 $（$ q $奇数）。作为原始$ 3N+1 $问题的自然概括，它由算术类型的离散动力系统组成。使用数字理论和动力学系统的标准方法，建立了一般属性，例如每个$ Q $有限的周期序列存在有限的周期序列。特别是，当$ q $是梅森（Mersenne）号码时，$ q = 2^p-1 $，只有一个这样的循环，即琐碎的周期。基于概率模型的进一步分析表明，对于$ q = 3 $，所有序列的渐近行为始终是收敛的，而对于$ q \ geq 5 $，序列的渐近行为对于几乎所有数字都有分歧（对于一组自然密度）。这得出的结论是，所谓的Collatz猜想是正确的，而$ Q = 3 $是其他人非常特殊的情况（Crandall猜想）。确实，猜想一般问题$ qn+1 $是不可决定的。
In this work the generalized Collatz problem $qn+1$ ($q$ odd) is studied. As a natural generalization of the original $3n+1$ problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number theory and dynamical systems, general properties are established, such as the existence of finitely many periodic sequences for each $q$. In particular, when $q$ is a Mersenne number, $q=2^p-1$, there only exists one such cycle, known as the trivial one. Further analysis based on a probabilistic model shows that for $q=3$ the asymptotic behavior of all sequences is always convergent, whereas for $q\geq 5$ the asymptotic behavior of the sequences is divergent for almost all numbers (for a set of natural density one). This leads to the conclusion that the so called Collatz Conjecture is true, and that $q=3$ is a very special case among the others (Crandall conjecture). Indeed, it is conjectured that the general problem $qn+1$ is undecidable.
