论文标题
没有$ M \ LEQ 91 $的Collatz-M-Cycles
There are no Collatz-m-Cycles with $m\leq 91$
论文作者
论文摘要
Collatz猜想（或``Syracuse问题''）考虑了递归定义的正整数序列，其中$ n $由$ \ tfrac {n} {2} $继承，如果$ n $均匀，或$ \ tfrac {$ \ tfrac {3n+1}} {2} {2} {2} {2} {$ n $是奇怪的。猜想指出，对于所有起始值$ n $，序列最终都达到了琐碎的周期$ 1、2、1、2，\ ldots $。我们对非平凡周期的存在感兴趣。 让$ m $成为这种非平凡周期中本地最小值的数量。 Simons和De Weger证明了$ M \ GEQ 76 $。在检查了Collatz猜想的启动值范围内，较新的界限，一个人获得了$ m \ geq 83 $。在本文中，我们证明了$ M \ GEQ 92 $。 本文的最后一部分考虑了必须证明的，以增加奇数成员的数量，必须将非平凡的周期纳入下一个界限 - 即至少$ k \ geq1.375 \ cdot 10^{11} $。我们证明，对于每个小于或等于$ 1536 \ cdot2^{60} = 3 \ cdot2^{69} $的整数，相应的collatz序列进入琐碎循环是足够的。这将要检查的数字范围减少了近60美元。
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 $60$\%.
