---

Statement

The prize problem (p. 4, unlabeled). The survey traces the problem to the contact between Klarner and Erdős at the University of Reading in 1971, which it says led to "a (solved) Erdős prize problem", posed as, quoted: "Does the smallest set $S_1$ of integers containing 1 and closed under the affine maps $x\mapsto2x+1$, $x\mapsto3x+1$ and $x\mapsto6x+1$ have a positive (lower asymptotic) density?" It reports the answer second-hand: $S_1$ was shown to have density zero by D. J. Crampin and A. J. W. Hilton, unpublished, according to Klarner (its [53], J. Algebra 74 (1982), 40--48); and the solvers collected the prize from Erdős (its [50], "A. J. W. Hilton, private communication, 2010").

Klarner's Integer Sequence Problem (p. 4, labeled), posed by Klarner ([53, p. 47]) as a revision, quoted: "Does the smallest set of integers $S_2$ containing 1 and closed under the affine maps $x\mapsto2x$, $x\mapsto3x+2$ and $x\mapsto6x+3$ have a positive (lower asymptotic) density?" The survey says that this problem remains unsolved and points to Guy's paper in the same volume (its [40]).

The survey places both problems beside the backward form of the $3x+1$ problem, the set $S_0$ generated from $1$ by $x\mapsto2x$ and $3x+2\mapsto2x+1$ (the conjecture page), as problems on sets of integers closed under affine maps.

Source. J. C. Lagarias, The $3x+1$ problem: an overview, in The Ultimate Challenge: The $3x+1$ Problem (AMS, 2010), 3--29; the arXiv:2111.02635v1 copy, p. 4, with references [40], [50] and [53] on pp. 25--26, read on the page images. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read clause by clause on the page image. The density-zero result is reported, not proved, and no written proof is named; Klarner's 1982 paper was not read here. Nothing here is independently reviewed.

Proof pointer

None in this survey. A reconstruction of Crampin and Hilton's argument, giving at most $C(\epsilon)T^{\tau_1+\epsilon}$ elements of $S_1$ up to $T$ with $\tau_1\approx0.900526$, is Theorem 6 of Lagarias 2016.

Dependencies

Klarner 1982 (the survey's [53]) and Hilton's 2010 private communication (its [50]), as reported.

Bears on

• Problem 1134: the survey's prize problem on $S_1$ is the problem's question, with the same three maps and positive lower density asked; the survey reports the negative answer second-hand (Crampin and Hilton, unpublished, according to Klarner) and gives no proof. Klarner's Integer Sequence Problem is a different question, with the maps $2x$, $3x+2$, $6x+3$, which the survey calls unsolved.
• Problem 1135: the passage is the survey's record of the link between Erdős and the circle of $3x+1$ problems through Klarner, which the problem page's Erdős-connection account cites; it says nothing about the conjecture itself.