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Statement

Section 2, "Numerical Investigations and Stopping Time" (pp. 3--12), its opening pp. 3--4 of the English version.

The trichotomy (p. 3). Every orbit of $T$ on the positive integers ends in one of three ways: it reaches the trivial cycle $\{1,2\}$, it reaches a nontrivial cycle, or it is divergent. The $3x+1$ problem asserts that the first always happens.

Verification (p. 3). Oliveira e Silva (the survey's [61], [62], 1999 and 2000) proved that the first alternative holds for every $n<100\times2^{50}\approx1.12\times10^{17}$, in a computation that ended in April 2000. Roosendaal ([65], 2003) is reported as claiming an extension to $n=195\times2^{50}\approx2.19\times10^{17}$; the survey reports this as a claim, not as a proof.

Cycle length (pp. 3--4). The survey reports as the record that a nontrivial cycle must have length no less than $272{,}500{,}658$, obtained from numerical verification bounds of the kind above together with continued fractions, and refers to its Section 5 for cycles. The passage names no source for the record; in Section 5, $272500658$ is the coefficient of $b$, with $b\ge1$, in Tempkin and Arteaga's length formula (see Section 5).

Stopping times (p. 4). The survey defines the stopping time $\sigma(n)=\inf\{k:T^{(k)}(n)<n\}$, the total stopping time $\sigma_\infty(n)=\inf\{k:T^{(k)}(n)=1\}$ and the height $h(n)=\sup\{T^{(k)}(n):k\in\mathbf Z^+\}$, with the example $\sigma(27)=59$, $\sigma_\infty(27)=111$, $h(27)=9232$; and (§ 2.1, p. 4) it restates the $3x+1$ problem as the claim that every positive integer has finite stopping time.

These are records as of 2003; the verification bound is now $2^{71}$ (Barina 2025) and the cycle-exclusion frontier is $m\le91$ local minima (Hercher 2023).

Source. M. Chamberland, An Update on the $3x+1$ Problem, author's English version of the survey in Butll. Soc. Catalana Mat. 18 (2003), 19--45; pp. 3--4 of the English version, 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 images. A survey's report of other authors' results; the cited sources were not read here.

Proof pointer

None here. Verification: Oliveira e Silva, Math. Comp. 68 (1999), 371--384 (the survey's [61]), and his web page ([62]); neither is held. Roosendaal's figure is a claim on his web page of records ([65]). Cycle length: Tempkin and Arteaga ([75], a 1997 draft, not held), as reported in Section 5 of the survey.

Dependencies

The cited papers, as reported.

Bears on

• Problem 1135: the 2003 status of the problem's question for its map $f$, which is the survey's $T$: no counterexample below $100\times2^{50}$, and no nontrivial cycle of length below $272{,}500{,}658$. These are partial results, superseded by the current values the problem page cites.