Chamberland 2003 update survey

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conjecture_p2: The 3x+1 conjecture as the survey poses it for the Collatz function C, with its compressed map T (the map of Problem 1135), the odd-to-odd map F, and the Erdős remark it quotes in its opening paragraph.

section_2: The survey's 2003 status of the 3x+1 problem for the map T: every orbit ends in the trivial cycle, a nontrivial cycle or divergence; Oliveira e Silva's verification for all n below 100 x 2^50, Roosendaal's claimed extension to 195 x 2^50, and the record that a nontrivial cycle has length at least 272,500,658.

section_2_3: The survey's account of lower bounds on Z_a(x), the number of positive integers up to x whose T-orbit reaches a: Crandall's existence of some c > 0 with Z_1(x) > x^c for large x, the successive exponents 0.25, 0.643, 3/7, 0.48 and 0.81, and Krasikov and Lagarias's Z_1(x) > x^0.84.

section_5: The survey's account of what is known about cycles of T: the author's identity between the even and odd terms of a cycle and his residue observations, the circuit theorem that {1,2} is the only circuit, the cycle-length forms of Eliahou and of Tempkin and Arteaga that give the bound 272,500,658, and Brox's finiteness theorem for cycles with few terms congruent to 1 mod 4.

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Marc Chamberland, An Update on the 3x+1 Problem, 2003 (author's English version of the survey published in Butll. Soc. Catalana Mat. 18 (2003) 19-45; 32 pp.).

Panoramic survey organized by attack surface: numerical investigations, stopping times and the Collatz graph, representations of iterates, reduction to residue classes, cycles, extensions of the map to the integers, rationals with odd denominators, the 2-adic and Gaussian integers, the real line and the complex plane, generalizations, and a miscellaneous section (a logical encoding of the problem and a global orbit statistic). It leaves out the work on functional equations, cellular automata and the origin of the problem, pointing to Wirsching's book for them (p. 3). Relevance: Survey of the 3x+1 problem organized by attack surface, the same survey genre the site cites (La85/La10/La16) for problem 1135.

Source: PDF. The copy read for this card is the author's English version, which prints no notice; the card records no hosting URL, so no publisher's or repository's record was read for it; the term is unstated.

Read status: claims checked for the definitions of $C$, $T$ and $F$, the conjecture and the Erdős remark (p. 2), the records of Section 2 (pp. 3--4), the predecessor-set bounds of § 2.3 (pp. 10--11) and the cycle results of Section 5 (pp. 15--17), read clause by clause on the page images; the rest of the survey was read for structure only. It is a survey: apart from the author's own cycle observations in Section 5 it reports other authors' results, whose sources were not read here.

Contents

• Section 1 (pp. 2--3), p. 2, [[number_theory/chamberland_2003_update_survey/conjecture_p2|the $3x+1$ conjecture]]: the Collatz function $C$, the conjecture, the compressed map $T$ (the problem page's $f$), the odd-to-odd map $F$, and the Erdős remark "Mathematics is not ready for such problems", quoted without a reference.
• Section 2 (pp. 3--12), its opening (pp. 3--4), the 2003 records: verification below $100\times2^{50}$, Roosendaal's claimed $195\times2^{50}$, nontrivial cycles of length at least $272{,}500{,}658$; stopping time, total stopping time and height.
• § 2.3 (pp. 9--12), predecessor-set bounds: $Z_1(x)>x^c$ for large $x$, from Crandall's $c>0$ to Krasikov and Lagarias's $c=0.84$ (p. 10).
• Section 5 (pp. 15--17), cycles: the author's cycle identity and residue observations, $\{1,2\}$ the only circuit, the cycle-length forms of Eliahou and of Tempkin and Arteaga, and Brox's finiteness result.

Bears on. #1135: the problem's map $f$ is the survey's $T$ and its question is the survey's conjecture stated for $T$; the survey records the conjecture and, as of 2003, partial results on it (verification below $100\times2^{50}$, no nontrivial cycle shorter than $272{,}500{,}658$, $Z_1(x)>x^{0.84}$), and proves nothing that settles it.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.