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Source. The unnumbered statement headed "Periodicity Conjecture", p. 3, 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).

The paper notes that $\Phi(\mathbf Q\cap\mathbf Z_2)\subseteq\mathbf Q\cap\mathbf Z_2$ is known, and easily proved from formula (1.6) (p. 2, citing [2]). It then records the following conjecture as proposed in its reference [8].

Periodicity Conjecture (p. 3). "$\Phi(\mathbf Q\cap\mathbf Z_2)=\mathbf Q\cap\mathbf Z_2$."

A trajectory $\{T^k(n):k\ge1\}$ is divergent when it contains infinitely many distinct elements, so that $|T^k(n)|\to\infty$ as $k\to\infty$ (p. 3). The paper states three consequences or equivalents, none proved in this paper:

• the conjecture would imply that $T$ has no divergent trajectories on $\mathbf Z$ (p. 3);
• with $T_{3,k}(x)=(3x+k)/2$ for odd $x$ and $x/2$ for even $x$, the conjecture is equivalent to the assertion that for all $k\equiv\pm1\pmod6$ the $3x+k$ function has no divergent trajectories on $\mathbf Z$, which the paper says follows from [9, Corollary 2.1b] (p. 3);
• for any $k\ge1$ it is equivalent to $\Phi^k(\mathbf Q\cap\mathbf Z_2)=\mathbf Q\cap\mathbf Z_2$ (p. 3).

The paper leaves the conjecture open.

Proof pointer

No proof: the conjecture is open. The equivalence with the $3x+k$ statement is credited to [9] (J. C. Lagarias, Acta Arith. 56 (1990), 33--53, Corollary 2.1b); the equivalence for $\Phi^k$ is noted on p. 3 without a written proof.

Dependencies

The definition of $\Phi$ and formula (1.6), p. 2.

Bears on

• Problem 1135: the problem's map $f$ is the paper's $T$ on the positive integers. By the paper's p. 3, the Periodicity Conjecture would imply that $T$ has no divergent trajectory on $\mathbf Z$, so every orbit of $f$ would be eventually periodic. That excludes divergent orbits only: it says nothing about cycles of $f$ other than $\{1,2\}$, so it would not by itself settle the problem. The conjecture is open, and the paper proves nothing toward it.