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Statement

As printed on p. 12 (Section 6, "Results"):

"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 $2\,048\times2^{60}$). This is the moment when the length of a non-trivial cycle rises to $355\,504\,839\,929$ [12]. See Table 10 for a timeline from the start of our project."

Table 10 ("Timeline of our project verifying the convergence of the Collatz conjecture") dates the project's start to 2019-09-04 and lists the verification bound reached on each later date: every number below $2^{68}$ by 2020-05-07, below $2^{69}$ by 2021-12-10, below $2^{70}$ by 2023-07-09, below $1.5\times2^{70}$ by 2023-11-03, and below $2^{71}$ by 2025-01-15.

"The Collatz conjecture" here is the assertion (p. 2) that repeated application of $T(n)=(3n+1)/2$ ($n$ odd), $n/2$ ($n$ even) "always converges to the cycle passing through the number 1 for arbitrary positive integer $n$"; $T$ is the map $f$ of Problem 1135, so the statement is that $f^{(k)}(m)=1$ for some $k$ whenever $m<2^{71}$. The cycle-length remark cites [12] (Eliahou's method of lower bounds on cycle lengths from the verification limit). A finite computation: it decides the conjecture for each $m$ below $2^{71}$ and nothing beyond.

Source. D. Barina, Improved verification limit for the convergence of the Collatz conjecture, J. Supercomput. 81 (2025), Article 810; Section 6 and Table 10 on p. 12 (PDF p. 12), display (1) on p. 2, read on the rendered page image of p. 12 and the text layer of p. 2. The artifact is identified in the source digest.

Read depth. Claims checked: the statement and Table 10 were read clause by clause on the page image. The computation was not rerun; the paper's own record of a distributed computation, refereed (accepted 21 April 2025), not replicated here.

Proof pointer

Sections 3--5 (pp. 3--12): the baseline algorithm of the author's 2020 paper (tracking the trajectory on $n$ and $n+1$ with trailing-zero counts and a small table of powers of $3$), $3^k$ sieves with optimized code for the $3^2$ sieve (Section 3.2), the $2^k$ sieve (Section 3.3), whose size $2^{34}$ is found optimal on the CPU in Section 5, the distribution of work units to thousands of parallel CPU and GPU processes on European supercomputers (Section 4), and the performance comparison (Section 5). The programs are released as open-source software (p. 13).

Dependencies

None mathematical; a computation whose correctness rests on the programs and the project's records.

Bears on

• Problem 1135: the current finite verification frontier, $2^{71}$, for the page's map $f$; it supersedes the $2^{68}$ of the author's 2020 paper and cannot settle the question for all $m$.