不存在 $m\leq 91$ 的 Collatz-m-循环
There are no Collatz-m-Cycles with $m\leq 91$
AI总结：
本文研究了Collatz猜想的非平凡循环，证明了局部最小值数量 $m \geq 92$，并指出验证 $3\cdot2^{69}$ 以内整数进入平凡循环可将奇数成员下界提至 $1.375\cdot 10^{11}$，减少近60%检查范围。
摘要：
Collatz 猜想（或称 Syracuse 问题）考虑正整数的递归定义序列，其中若 $n$ 为偶数，则 $n$ 后继为 $\tfrac{n}{2}$；若 $n$ 为奇数，则后继为 $\tfrac{3n+1}{2}$。该猜想指出，对于所有起始值 $n$，序列最终会到达平凡循环 $1, 2, 1, 2, \ldots$。我们关注非平凡循环的存在性。设 $m$ 为此类非平凡循环中局部最小值的数量。Simons 和 de Weger 证明了 $m \geq 76$。利用 Collatz 猜想已验证起始值范围的较新界限，可得 $m \geq 83$。本文证明了 $m \geq 92$。本文最后部分考虑了为将非平凡循环必须具有的奇数成员数量提升至下一界限（即至少 $K\geq1.375\cdot 10^{11}$）所需证明的内容。我们证明，只需证明对于每个小于或等于 $1536\cdot2^{60}=3\cdot2^{69}$ 的整数，相应的 Collatz 序列进入平凡循环即可。这将待检查数字的范围减少了近 $60$\\%。
英文摘要：
The Collatz conjecture (or ``Syracuse problem'') considers recursively-defined sequences of positive integers where $n$ is succeeded by $\tfrac{n}{2}$, if $n$ is even, or $\tfrac{3n+1}{2}$, if $n$ is odd. The conjecture states that for all starting values $n$ the sequence eventually reaches the trivial cycle $1, 2, 1, 2, \ldots$ . We are interested in the existence of nontrivial cycles. Let $m$ be the number of local minima in such a nontrivial cycle. Simons and de Weger proved that $m \geq 76$. With newer bounds on the range of starting values for which the Collatz conjecture has been checked, one gets $m \geq 83$. In this paper, we prove $m \geq 92$. The last part of this paper considers what must be proven in order to raise the number of odd members a nontrivial cycle has to have to the next bound -- that is, to at least $K\geq1.375\cdot 10^{11}$. We prove that it suffices to show that, for every integer smaller than or equal to $1536\cdot2^{60}=3\cdot2^{69}$, the respective Collatz sequence enters the trivial cycle. This reduces the range of numbers to be checked by nearly $60$\%.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
