[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
