Barina 2025 improved verification limit convergence collatz

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section_6: Barina's 2025 result statement that the distributed project verified the convergence of the Collatz conjecture for every starting value up to 2^71, with the project timeline dating the 2^71 milestone to 15 January 2025; the current finite verification record for Problem 1135.

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David Barina, Improved verification limit for the convergence of the Collatz conjecture, J. Supercomput. 81 (2025), no. 7, Article 810, 14 pp.; DOI 10.1007/s11227-025-07337-0. "Accepted: 21 April 2025" (p. 1); published online 2 May 2025 (the Crossref record); an open-access article ("© The Author(s) 2025"). Not cited by the site's Problem 1135 page.

The retained folder-name PDF is the publisher's PDF, 14 pages with a complete text layer, printed as "810, Page $n$ of 14"; the statement pages were read on the rendered page images of pp. 1 and 12. Provenance: retained from the repository's survey download set of 5 September 2026 (the download URL was not recorded; the identifier is the DOI https://doi.org/10.1007/s11227-025-07337-0 printed on the first page); 862,781 bytes. The file prints "© The Author(s) 2025" on its first page and "Open Access This article is licensed under a Creative Commons Attribution 4.0 International License" on p. 13: the Creative Commons Attribution 4.0 license.

Read status: claims checked for the abstract, the introduction's definition of the map $T$ and its status sentences, the related-work list, the Section 6 result statement and Table 10 (the project timeline), and the conclusion, read clause by clause on the page images of pp. 1 and 12 and the text layer of pp. 2--3 and 13--14; the algorithms (Sections 3--5) were read for structure only.

Contents

• Abstract and Section 1 (pp. 1--2): the project "aims to verify the Collatz conjecture computationally"; "a new result that pushes the limit for which the conjecture is verified up to $2^{71}$"; the map $T(n)=(3n+1)/2$ ($n$ odd), $n/2$ ($n$ even) (display (1), p. 2), replacing the $3n+1$ branch by $(3n+1)/2$ since the former's output is even; "The conjecture has never been proven"; "At the time of writing this article, all starting values up to $2^{68}$ were computer-checked [4]. This paper presents a result that pushes this limit to $2^{71}$."
• Section 2 (pp. 2--3): earlier records (Dunn 1973, about $2^{24.78}$; Leavens--Vermeulen 1992, about $2^{45.67}$; Roosendaal $2^{60}$; Oliveira e Silva $5\times2^{60}$; yoyo@home 2017, $87\times2^{60}$; the author's project 2019--2021, $2^{68}$ [4]).
• Sections 3--5 (pp. 3--12): the baseline algorithm, $3^k$ sieves and the $2^{34}$ sieve, the distributed architecture on European supercomputers, performance tables (the total speedup $1335.9\times$ from the first CPU algorithm to the best GPU algorithm).
• Section 6, Results (p. 12): "At the time of writing this article, we have managed to verify the convergence of the Collatz conjecture for all numbers up to the limit of $2^{71}$ (which is equal to $2048\times2^{60}$). This is the moment when the length of a non-trivial cycle rises to $355\,504\,839\,929$ [12]." Table 10, the project timeline: started 2019-09-04; all numbers below $2^{68}$ verified; below $2^{69}$ 2021-12-10; below $2^{70}$ 2023-07-09; below $1.5\times2^{70}$ 2023-11-03; below $2^{71}$ 2025-01-15. Five new path records (OEIS A006884) found during the verification (pp. 12--13); the result statement.
• Section 7 (p. 13): conclusion; open-source release of the programs.

Compiled scope

The statements were read; the Section 6 result 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 current finite verification record for the page's map $f=T$, every starting value below $2^{71}$ (15 January 2025), refereed; it replaces the $2^{68}$ record of the author's 2020 paper and decides nothing beyond $2^{71}$.

Results.

• Section 6, Results (p. 12): the convergence of the Collatz conjecture is verified for all numbers up to $2^{71}=2048\times2^{60}$; the $2^{71}$ milestone dated 2025-01-15 in Table 10.