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Statement

Setting (p. 1). $\mathbb Z_2$ is the ring of 2-adic integers. Every $Q\in\mathbb Z_2$ has a unique expansion $Q=2^{d_0}+2^{d_1}+\cdots$ (display (1)) over an increasing, finite or infinite, sequence $0\le d_0<d_1<\cdots$ of nonnegative integers, and every $N\in\mathbb Z_2$ has a unique expansion $N=\frac{-1}{3}2^{d_0}+\frac{-1}{9}2^{d_1}+\frac{-1}{27}2^{d_2}+\cdots$ (display (2)) over such a sequence. Matching the two expansions through a common sequence $d=\langle d_0,d_1,\ldots\rangle$ defines a bijection $\Phi$ of $\mathbb Z_2$, written $N=\Phi(Q)$. Under (1) the finite sequences correspond to the nonnegative integers $Q$, the empty sequence to $Q=0$. The paper derives both bijections from the general fact that, for fixed odd $u_0,u_1,\ldots\in1+2\mathbb Z_2$, the map from increasing sequences to sums $\sum u_i2^{d_i}$ is one-to-one and onto $\mathbb Z_2$.

Example (display (3), p. 1). $\Phi(-1/3)=\Phi(2^0+2^2+2^4+\cdots)=1$, so $1\in\Phi((1/3)\mathbb Z)$.

Conjecture (p. 1, unnumbered): "The set $\mathbf{Z}^+$ of positive integers is contained in $\Phi((1/3)\mathbf{Z})$."

The abstract (p. 1) states the same conjecture in the form that $3Q$ is an integer whenever $\Phi(Q)$ is a positive integer.

Equivalence with the 3N+1 conjecture

Theorem 2 and Theorem 3 (p. 2) show, integer by integer, that a positive integer $N$ lies in $\Phi((1/3)\mathbb Z)$ exactly when some iterate of the Collatz map $C$ sends it to $1$. The paper concludes (p. 2) that the conjecture is equivalent to the $3N+1$ conjecture. The conjecture is the paper's non-iterative statement: it is phrased through the expansions (1) and (2), with no iteration of $C$.

Read depth

Claims checked: the definition of $\Phi$, the example and the conjecture were read clause by clause on the page images of the print. A second reader checked the definition, the example, the statement, label and page against the print.

Dependencies

None.

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: the conjecture is equivalent to the $3N+1$ conjecture by Theorems 2 and 3, and the problem page records that a $C$-orbit reaches $1$ exactly when the orbit of the problem's shortcut map $f$ does, so the conjecture is a restatement of the problem's question. The paper proves only the equivalence, not the conjecture.