✕
从 5+6x 到 8+6x：考拉兹迭代中一类数字的局部收敛定理
From 5+6x to 8+6x: A Local Convergence Theorem in the Collatz Iteration
V1
摘要
[目的] 研究考拉兹迭代中形如 5+6x 的数在复合运算 T (n)=(3n+1)/2 下的局部收敛行为。
[方法] 将迭代过程中出现的所有奇数统一表示为算术级数 S (x)=a+bx，采用归纳法证明系数公式 a＿k = 2×3^(k+1)-1、b＿k = 2×3^(k+1)，建立迭代继续条件与初始参数二进制表示的对应关系，利用二进制表示的有限性完成终止性证明。
[结果] 证明了从任意 5+6x 出发，经有限次 T 迭代后必然落入偶数数列 8+6x，并给出收敛步数上界 K+1=L+2（L 为初始参数的最高二进制位）。
[局限] 仅研究模 6 余 5 这一类数，且仅在复合运算 T=(3n+1)/2 下讨论，不处理偶数后续除以 2 的步骤，结论是局部的，不构成对完整考拉兹猜想的证明。
[结论] 为考拉兹猜想中模 6 余 5 数类的迭代行为提供了完整且严格的局部描述。
Abstract
[Objective] To study the local convergence behavior of numbers of the form 5+6x under the composite operation T(n)=(3n+1)/2 in the Collatz iteration. [Methods] All odd numbers arising in the iteration are uniformly represented as an arithmetic progression S(x)=a+bx. The closed-form coefficients a＿k = 2×3^(k+1)-1、b＿k = 2×3^(k+1) are established by induction. A precise correspondence between the iteration continuation condition and the binary representation of the initial parameter is established, and the termination proof is completed using the finiteness of the binary representation. [Results] It is proved that starting from any number of the form 5+6x, after finitely many iterations of T, one necessarily reaches an even number belonging to the sequence 8+6x. An upper bound on the number of convergence steps is given by K+1=L+2, where L is the highest binary bit of the initial parameter. [Limitations] Only the class of numbers congruent to 5 modulo 6 is studied, and only under the composite operation T=(3n+1)/2. The subsequent division-by-2 steps for even numbers are not addressed. The conclusion is local and does not constitute a proof of the full Collatz conjecture. [Conclusion] A complete and rigorous local description is provided for the iterative behavior of the class of numbers congruent to 5 modulo 6 in the Collatz conjecture.关键词
考拉兹猜想/3n+1问题/二进制表示/归纳证明/局部定理Key words
Collatz conjecture/ 3n+1 problem/ binary representation/ proof by induction/ local theorem引用本文复制引用
郝占柱.从 5+6x 到 8+6x：考拉兹迭代中一类数字的局部收敛定理[EB/OL].(2026-09-30)[2026-10-30].https://sinoxiv.napstic.cn/article/26282078.学科分类
数学版本历史
| 版本 | 时间 | DOI号 | 操作 |
|---|---|---|---|
| v2 | 2026-10-29 16:15:15 | 10.12383/202609050002V2 | 全文下载 |
| v1（当前版本） | 2026-09-30 15:58:00 | 10.12383/202609050002V1 | 全文下载 |