Submitted:
02 October 2024
Posted:
02 October 2024
Read the latest preprint version here
Abstract
Research Collatz odd sequence, change (×3 + 1) ÷ 2k operation in Collatz Conjecture to (×3 + 2m − 1) ÷ 2k operation. Expand loop Collatz odd sequence(if exists) in (×3 + 2m − 1) ÷ 2k odd sequence to become ∞ steps non-loop sequence. Build (×3 + 2m − 1) ÷ 2k odd tree model and transform position model for odds in tree.Via comparing actual and virtual positions, prove if a (×3+2m −1)÷2k odd sequence can not converge after infinite steps of (×3+2m −1)÷2k operation, the sequence must walk out of boundary of the tree. Terms: (×3 + 1) ÷ 2k: ∀b ∈ O{1, 3, 5, 7, 9...}, O{...} is odd set, do b × 3 + 1, then do k times ÷2 repeatedly until get an odd. At this point, these odds are called Collatz odds. (×3 + 2m − 1) ÷ 2k: ∀a ∈ O{3, 5, 7, 9...}, highest binary bit is 2m−1, do a × 3 + 2m − 1, then do k times ÷2 repeatedly until get an odd.
