句柄、钩子和场景：对Collatz猜想的新认识
Handles, Hooks, and Scenarios: A fresh Look at the Collatz Conjecture
摘要：
给出了Collatz猜想的一种运算方法。场景被定义为字符串“s”（代表“spike”）和“d”（代表“down”），它们象征着连接两个奇数（起始数和结束数）的Collatz级数中的Collatz运算（3 m +1）/2和m/2。它表明，一个场景确定唯一的四个整数，称为startperiod，startphase，endperiod，和endphase，使startnumber = startperiod x k减去startphase，和endnumber = endperiod x k减去endphase，其中k是任何自然数。因此，任何场景都可以实现无限次，每次实现都有明确定义的startnumber和endnumber。它显示了如何计算给定情景的期间和阶段。这些结果被用来证明Collatz级数中的任何奇（偶）数与可被3整除的奇数的距离小于8（7）（“上”或“下”）步。它们也用于Collatz系列的构建，这些系列表现出规定的规则图形模式。最后，讨论了目前工作对非平凡圈问题的意义。
英文摘要：
An operational approach to the Collatz Conjecture is presented. Scenarios are defined as strings of characters "s" (for "spike") and "d" (for "down") which symbolize the Collatz operations (3m+1)/2 and m/2 in a Collatz Series connecting two odd integers, the startnumber and the endnumber. It is shown that a scenario determines uniquely four integers, called startperiod, startphase, endperiod, and endphase such that startnumber = startperiod x k minus startphase, and endnumber = endperiod x k minus endphase, where k is any natural number. Therefore, any scenario can be realized infinitely many times, with well-defined startnumber and endnumber for each realization. It is shown how the periods and phases for a given scenario are calculated. The results are used to prove that any odd (even) number in a Collatz Series is less than 8 (7) ("up"- or "down"-) steps away from an odd integer which is divisible by 3. They are also used for the construction of Collatz Series which exhibit prescribed regular graphics patterns. Finally, the bearing of the present work on the question of non-trivial cycles is discussed.
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