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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; differentiability is with respect to the 2-adic metric.

Corollary 2 (p. 3): "$\Phi$ is nowhere differentiable."

The paper credits the theorem to Müller (H. Müller, Das 3n+1 Problem, Mitteilungen der Math. Ges. Hamburg 12 (1991), 231--251) and gives a short proof; the proof's last line also concludes that $\Phi^{-1}$ is nowhere differentiable. The same section notes (p. 3) that by (1) and (2) $\Phi$ is a homeomorphism of $\mathbb Z_2$.

Proof pointer

Difference quotients at $Q$ are computed from (2) (p. 3). If the exponent sequence $d$ of $Q$ is infinite, the quotient for the increment $-2^{d_k}$ is congruent to $(-1)^k$ modulo $4$, so it has no limit as $k\to\infty$. If $d$ is finite, of length $m$, the quotient for the increment $2^e+2^{e+f}$ with $e>d_{m-1}$ equals $3^{-(m+2)}(3+2^f)/(1+2^f)$, which takes different values modulo $8$ for $f=1$ and $f=2$, so it has no limit as $e\to\infty$.

Read depth

Claims checked: the statement was read on the page images of the print, and the proof was followed; the two routine computations the paper leaves to the reader were not redone. 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 describes the 2-adic map that conjugates $H$ to the Collatz map; it says nothing about whether orbits of positive integers reach $1$.