Barina 2020 convergence verification

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verification_p6: The unnumbered statement of Barina's 2020 paper that the distributed project running its algorithm verified the Collatz conjecture for all starting values below 2^68 between September 2019 and May 2020; a finite computational record for Problem 1135, since raised to 2^71 by the same project.

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David Barina, Convergence verification of the Collatz problem, J. Supercomput. 77 (2021), no. 3, 2681--2688; DOI 10.1007/s11227-020-03368-x (published online 1 July 2020; the Crossref record). The copy read for this card is the author's postprint from the Brno University of Technology publication repository, 8 pages with its own pagination ("Received: date / Accepted: date" on p. 1), complete text layer; the journal text was not compared; the locators below are the postprint's pages. The statement pages were read on the rendered page images of pp. 6--7. Source: PDF. That postprint comes from the Brno University of Technology repository, for which no hosting statement is recorded, and it prints no notice; the publisher's page for the version of record (DOI 10.1007/s11227-020-03368-x) shows "© Springer Science+Business Media, LLC, part of Springer Nature 2020" and no Creative Commons or Open Access statement, which governs the article as published and not this postprint; the term is unstated.

Read status: claims checked for the abstract, the introduction's definition of the map and its status sentences, and the verification statement of Section 5 with the path-record remark (pp. 1, 6--7), read clause by clause on the page images of pp. 6--7 and the text layer of p. 1; the algorithm (Sections 3--4) and the performance table were read for structure only.

Presents the verification algorithm — small O(N) lookup tables in place of O(2^N) precomputed tables — behind the record computational check that every starting value below 2^68 converges (the distributed project it reports has since pushed the checked floor onward: the author's 2025 paper reports $2^{71}$). Relevance: Reports and describes the computational verification that every starting value below 2^68 converges under the Collatz map (problem 1135).

Contents

• Section 1 (pp. 1--2): the Collatz function $C(n)=3n+1$ ($n$ odd), $n/2$ ($n$ even) and the conjecture that iteration "will always converge to the cycle passing through the number 1"; "The conjecture has never been proven"; as of 2020 checked for all starting values up to $10^{20}$ [1]; the idea of tracking the trajectory on $n+1$ and switching between the $n$ and $n+1$ domains so that only multiplicative operations and the count of trailing zeros are used.
• Section 2 (pp. 2--3): related projects (Oliveira e Silva's $19\times2^{58}$ in 2008, Leavens--Vermeulen 1992, Dunn 1973, Roosendaal, yoyo@home, Honda et al.).
• Sections 3--4 (pp. 3--6): the algorithm (Algorithm 1) and its optimizations (sieves, congruence classes).
• Section 5, Performance Evaluation (pp. 6--7): Table 1 (speeds; the program verifies $2^{40}$ 128-bit numbers per work unit); the verification statement: "The program presented in this paper runs as a part of a distributed computing project to check the convergence of the Collatz problem. From September 2019 to May 2020, the project managed to verify this conjecture for all numbers below $2^{68}$." The path-record remark (p. 7): the records found up to $2^{68}$ "confirm" the Lagarias--Weiss prediction $\limsup\log t(n)/\log n=2$, in the paper's word, and the largest known path record below $2^{68}$ occurs for $n=274133054632352106267$.
• Section 6 (p. 7): conclusion; the open-source release of the programs.

Compiled scope

The statements were read; the verification statement is compiled as a page. The computation was not rerun and nothing here is independently reviewed; a computer verification is finite evidence, not a proof of the conjecture.

Bears on. #1135: the paper reports that a distributed computation (September 2019 to May 2020) verified the conjecture for all starting values below $2^{68}$, which answers the question yes for each $m<2^{68}$ and says nothing about larger $m$; the same project's later bound $2^{71}$ is recorded in the author's 2025 paper. Finite verification cannot settle the question for all $m$.

Results.

• Verification statement (pp. 6--7): all starting values below $2^{68}$ converge to the cycle through $1$ (a distributed computation, September 2019 to May 2020).

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