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Statement

Pp. 14--15 of the arXiv:2111.02635v1 preprint (Section 6, "Current Status", 6.1 "Where does research currently stand on the $3x+1$ problem?"). § 6.1 opens by calling the problem unsolved and a solution out of reach at present, and goes on, quoted (p. 14): "To quote a still valid dictum of Paul Erdős ([58, p. 3]) on the problem: 'Mathematics is not yet ready for such problems.'" It then lists five "world records", all resting on large computer calculations together with theoretical work:

• (W1) the $3x+1$ conjecture is verified for every $n<20\times2^{58}\approx5.7646\times10^{18}$ (Oliveira e Silva [76], in the same volume);
• (W2) on the positive integers, the only cycle of $T$ of period length less than $10{,}439{,}860{,}591$ is the trivial cycle $\{1,2\}$, which is also the only cycle with fewer than $6{,}586{,}818{,}670$ odd elements (Eliahou [24, Theorem 3.2]);
• (W3) infinitely many positive integers $n$ need at least $6.143\log n$ iterations of $T$ to reach $1$ (Applegate and Lagarias [3]);
• (W4) writing the number of iterations of $T$ that take $n$ to $1$ as $C\log n$, the largest known value of $C$ is $C\approx36.7169$, attained at $n=7{,}219{,}136{,}416{,}377{,}236{,}271{,}195$ (Roosendaal [79, $3x+1$ Completeness and Gamma records]);
• (W5) for all sufficiently large $X$, at least $X^{0.84}$ of the integers $1\le n\le X$ iterate to $1$ (Krasikov and Lagarias [57]).

A footnote to the Eliahou citation in (W2) (p. 15) identifies the cited bound as the bound $(21,0)$ of [24, Table 2] and says that the computations of (W1) now rule out the smaller values in that table. After the list the survey points to Brox [8] and to Simons and de Weger [83] for progress on excluding various kinds of periodic points of $T$, with bounds that rest on Diophantine approximation. Here $T(x)=(3x+1)/2$ for odd $x$ and $x/2$ for even $x$ (p. 1), the problem page's $f$; the cycle of (W2) is the cycle $\{1,2\}$ of $T$.

These are records as of 2010; the verification bound is now $2^{71}$ (Barina 2025) and the cycle-exclusion frontier is $m\le91$ local minima (Hercher 2023).

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, pp. 14--15 (PDF pp. 14--15), read on the rendered page images; [58] is J. C. Lagarias, The $3x+1$ problem and its generalizations, Amer. Math. Monthly 92 (1985), 3--23 (p. 26 of the reference list). The edition read is identified in the source digest.

Read depth. Claims checked: the passage was read clause by clause on the page images. A survey's report of other authors' results; the sources [3], [24], [57], [76], [79] were not read here except Krasikov--Lagarias's abstract (that paper has its own source card, which holds no file).

Proof pointer

None here. (W1): Oliveira e Silva's chapter in the same volume (not held). (W2): Eliahou, Discrete Math. 118 (1993), 45--56 (not held). (W3): Applegate and Lagarias, Math. Comp. 72 (2003), 1035--1049 (not held). (W5): Krasikov and Lagarias, Acta Arith. 109 (2003), 237--258, whose source card is krasikov_lagarias_2003_bounds_difference_inequalities (no file held; its abstract states that for each fixed positive integer $a$ not divisible by $3$ and all large enough $x$, at least $x^{0.84}$ of the integers below $x$ have $a$ in their forward orbit, which for $a=1$ is (W5); not consumed beyond that).

Dependencies

The cited papers, as reported.

Bears on

• Problem 1135: the 2010 status the site points to, with the Erdős dictum the site quotes from Guy in the form "Mathematics may not be ready for such problems" (Lagarias's wording, from the 1985 survey, has "is not yet" where Guy has "may not be"); the density bound $X^{0.84}$ (W5) is the strongest unconditional lower bound on the count of convergent starting values recorded on the page.