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Statement

As printed in the last paragraph of Section 5 (pp. 6--7 of the postprint):

"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 $2^{68}$."

Here "this conjecture" is the Collatz conjecture as the paper states it (p. 1): for every positive integer $n$, repeated application of $C(n)=3n+1$ ($n$ odd), $n/2$ ($n$ even) "will always converge to the cycle passing through the number 1". For the problem page's shortcut map $f$ the statement is the same, since the $f$-orbit of $n$ reaches $1$ exactly when the $C$-orbit does (the shortcut orbit omits only the even values $3n+1$ that follow odd terms, and $1$ is odd). The verification is a computation: it decides the conjecture for each $m<2^{68}$ and says nothing about larger $m$. The paper also records (p. 7) the largest path record found below $2^{68}$, $n=274133054632352106267$.

Source. D. Barina, Convergence verification of the Collatz problem, J. Supercomput. 77 (2021), no. 3, 2681--2688; the author's postprint, pp. 6--7 (PDF pp. 6--7), read on the rendered page images. The edition is identified in the source digest.

Read depth. Claims checked: the statement and its context were read clause by clause on the page images; the algorithm was read for structure and the computation was not rerun. The paper's own record of a computation carried out by a distributed project; no independent replication here.

Proof pointer

Sections 3--4 (pp. 3--6): Algorithm 1 tracks the trajectory alternately on $n$ and $n+1$ using only the count of trailing zeros, right shifts and a small table of powers of $3$, so that $N$ steps need a table of size $O(N)$ instead of $O(2^N)$; sieves on residue classes modulo $2^k$ skip the starting values that drop below themselves or join the trajectory of a smaller number within $k$ steps. Section 5 (p. 6): 128-bit arithmetic with a multi-precision fallback; work units of $2^{40}$ numbers, with partial (per-unit) path records stored and the sum of all the $\alpha$'s of Algorithm 1 over the unit kept "as proof of work so that the results can be independently verified". The programs were released as open-source software (p. 6, footnote 5).

Dependencies

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

Bears on

• Problem 1135: the finite verification frontier of 2020, $2^{68}$; the same project's 2025 paper reports $2^{71}$ (its Section 6). Finite computation cannot answer the question for every $m$.