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Statement

Setting (p. 1). $\Phi$ is the permutation of the 2-adic integers $\mathbb Z_2$ defined on the conjecture's page.

Corollary 1 (p. 3): "$\Phi(\mathbf{Q}\cap\mathbf{Z}_2)\subseteq\mathbf{Q}\cap\mathbf{Z}_2$."

The paper notes (p. 3) that the "Periodicity Conjecture" of Lagarias's 1985 survey asserts the equality $\Phi(\mathbb Q\cap\mathbb Z_2)=\mathbb Q\cap\mathbb Z_2$, and that the corollary is one half of it. The reverse inclusion is not proved.

Proof pointer

The argument precedes the statement (p. 3). For rational $Q$ the binary expansion (1) is finite or eventually periodic, so the exponent sequence $d$ is finite, of length $\mu$, or satisfies $d_{m+\lambda}=d_m+X$ for all $m\ge\mu$ and some fixed $\lambda$ and $X$. In the first case $3^\mu N$ is an integer; in the second, summing the geometric tail of (2) gives an explicit linear relation with integer coefficients showing $-3^\mu(3^\lambda-2^X)N$ is an integer. Either way $N=\Phi(Q)$ is rational.

Read depth

Claims checked: the statement and the argument before it were read on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print; the proof was not independently reviewed.

Dependencies

The definition of $\Phi$ (p. 1).

Source. Daniel J. Bernstein, A non-iterative 2-adic statement of the $3N+1$ conjecture, Proceedings of the American Mathematical Society 121 (1994), 405--408. Pages are those of the author's typescript named on the source card, numbered 1--4 rather than by the journal's pagination.

Bears on

• Problem 1135: background only. The corollary concerns the 2-adic conjugacy of the Collatz map and half of a conjecture from Lagarias's survey; it does not bear on whether orbits of positive integers reach $1$.