structural:summarization
2-adic extension

The function T (x) = { x 2 if x ≡ 0 (mod 2) 3 x + 1 2 if x ≡ 1 (mod 2) {\displaystyle T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}} is well-defined on the ring Z 2 {\displaystyle \mathbb {Z} _{2}} of 2-adic integers, where it is continuous and measure-preserving with respect to the 2-adic measure. Moreover, its dynamics is known to be ergodic.

Define the parity vector function Q acting on Z 2 {\displaystyle \mathbb {Z} _{2}} as Q (x) = ∑ k = 0 ∞ (T k (x) mod 2) 2 k. {\displaystyle Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.}

The function Q is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in Z 2 {\displaystyle \mathbb {Z} _{2}}.

An equivalent formulation of the Collatz conjecture is that Q (Z +) ⊂ 1 3 Z. {\displaystyle Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z}.}