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Source. The unnumbered statement headed "3x + 1 Conjecture", p. 2, of Daniel J. Bernstein and Jeffrey C. Lagarias, The 3x+1 conjugacy map, Canad. J. Math. 48 (1996) 1154--1169, with label and page as printed in the authors' retypeset manuscript dated 15 February 1996, the edition read for the source card.

Statement

Throughout, $T$ is the $3x+1$ function, $T(x)=(3x+1)/2$ for $x\equiv1\pmod2$ and $T(x)=x/2$ for $x\equiv0\pmod2$ (1.1), and $S$ is the 2-adic shift, $S(x)=(x-1)/2$ for odd $x$ and $S(x)=x/2$ for even $x$ (1.2), both on the 2-adic integers $\mathbf Z_2$ (p. 1). The $3x+1$ conjugacy map $\Phi$ is the unique map $\mathbf Z_2\to\mathbf Z_2$ with $\Phi\circ S\circ\Phi^{-1}=T$ and $\Phi(0)=0$ (pp. 1--2). It is a solenoidal bijection: $x\equiv y\pmod{2^n}$ implies $\Phi(x)\equiv\Phi(y)\pmod{2^n}$ for every $n$ (p. 2), so it induces a permutation $\Phi_n$ of $\mathbf Z/2^n\mathbf Z$ (p. 3).

For $x\in\mathbf Z_2$ written as $x=\sum_l2^{d_l}$ with $0\le d_1<d_2<\cdots$ a finite or infinite sequence, the paper records the explicit formula $\Phi(x)=-\sum_l3^{-l}2^{d_l}$ (1.6), citing [2] (p. 2). Here $\mathbf Z^+$ is the set of positive integers and $\tfrac13\mathbf Z=\{m/3:m\in\mathbf Z\}$, read inside $\mathbf Z_2$.

On p. 1 the paper states the $3x+1$ Conjecture as: for each positive integer $n$ some iterate $T^k(n)$ equals $1$, that is, all orbits on the positive integers reach the cycle $\{1,2\}$. On p. 2 it says that this conjecture is reformulated, citing its references [2] and [8], as

$3x+1$ Conjecture (p. 2). "$\mathbf Z^+\subseteq\Phi(\tfrac13\mathbf Z)$."

The paper gives the reformulation as known from those references and proves nothing about it; it is a restatement, not a result of this paper.

Proof pointer

None in this paper: the equivalence is attributed to [2] (D. J. Bernstein, Proc. Amer. Math. Soc. 121 (1994), 405--408) and [8] (J. C. Lagarias, Amer. Math. Monthly 92 (1985), 3--23).

Dependencies

The definition of $\Phi$ by (1.3) and $\Phi(0)=0$, and formula (1.6), both on p. 2.

Bears on

• Problem 1135: the problem's map $f$ is the paper's $T$ restricted to the positive integers, and its question, whether every $m\ge1$ has $f^{(k)}(m)=1$ for some $k\ge1$, is the $3x+1$ Conjecture as the paper states it on p. 1. The paper records, from its references [2] and [8], that this conjecture is equivalent to $\mathbf Z^+\subseteq\Phi(\tfrac13\mathbf Z)$. It is a restatement only; the paper does not prove or disprove either form, and says its own results "are not related to the 3x + 1 Conjecture in any immediate way" (p. 3).