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Collatz conjecture for the map T

Determine whether every positive integer n, under iteration of the Collatz map T defined by T(n) = (3n+1)/2 if n is odd and T(n) = n/2 if n is even, eventually reaches the 2-cycle (1,2).

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Background

The paper studies properties of Collatz iterations under the accelerated map T(n) = (3n+1)/2 for odd n and T(n) = n/2 for even n. The classical Collatz conjecture asserts eventual convergence to the trivial cycle (1,2) for all starting values.

This conjecture underpins much of the paper's motivation: paradoxical sequences are analyzed in part because their finiteness (above 2) would imply the Collatz conjecture, linking the paper’s main theme directly to this longstanding open question.

References

According to the Collatz conjecture, whatever the positive integer $n$ taken as first term, iterating the function

\begin{equation}\label{eq:T} T(n) = \left{\begin{array}{ll} \frac{3n+1}{2} & \mbox{if $n$ is odd,} \ \frac{n}{2} & \mbox{if $n$ is even,} \end{array}\right. \end{equation}

gives rise to a sequence $n, T(n), T{2}(n), \ldots$ which invariably reaches the trivial cycle $(1,2)$.

eq:T:

$T(n) = \left\{\begin{array}{ll} \frac{3n+1}{2} & \mbox{if $n$ is odd,} \\ \frac{n}{2} & \mbox{if $n$ is even,} \end{array}\right. $

— Paradoxical behavior in Collatz sequences (2502.00948 - Rozier et al., 2 Feb 2025) in Section 1 (Introduction)