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This page from 2017 by Eric Roosendaal says that the yoyo@home project checked for convergence all numbers up to approx. 266. Is it still a valid record? I am aware of the ongoing BOINC project, but I cannot find how far they are.

The same question from 2014: For how many consecutive numbers Collatz conjecture was checked?

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    $\begingroup$ This lists the record as of last month: ericr.nl/wondrous/progress.html $\endgroup$ – David G. Stork Aug 5 '19 at 17:51
  • $\begingroup$ boinc.thesonntags.com/collatz/highest_steps.php this was three clicks away from the BOINC homepage $\endgroup$ – Bananach Aug 5 '19 at 17:51
  • $\begingroup$ @DavidG.Stork If I understand this page correctly, then the project verified all numbers up to $509\,040 \times 20 \times 10^{12}$, which corresponds to approx. $2^{63}$. This is less than $2^{66}$. How does this project relate to the original page by Eric Roosendaal claiming that checked all numbers below $2^{66}$? $\endgroup$ – DaBler Aug 6 '19 at 11:47
  • $\begingroup$ @Bananach This page just lists the numbers having the highest number of steps so far. It says nothing about how much the problem was verified. Or did I misunderstand something? $\endgroup$ – DaBler Aug 6 '19 at 11:54
  • $\begingroup$ This (in German) seems to be an article explaining the Collatz sub-project. Would anyone be able to translate it for me? $\endgroup$ – DaBler Aug 6 '19 at 16:15
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Since nobody provided an answer to my question, I will answer myself.

  • As of August 2019, I am aware of ongoing BOINC project [1]. By personal correspondence with Eric Roosendaal I found that this ongoing BOINC project is meant to disprove the Collatz conjecture by trying to find a counter-example. The project started off in the middle of nowhere, at $2^{71}$ apparently, without specifying any arguments why this was chosen or why this would be a sensible point to use. It looks like they have reached roughly $2^{72.3}$ or so. No info is given as to whether all numbers up to that limit have indeed be checked.

  • As of August 2019, I am also aware of another ongoing project [2] by Eric Roosendaal. All numbers up to $2^{60} \approx 10^{18}$ have been checked for convergence.

  • In 2017, the yoyo@home project [3] [4] checked for convergence all numbers up to $10^{20} \approx 2^{66.4}$.

  • The paper by Tomás Oliveira e Silva [5] from 2010 claims that the author verified the conjecture up to $2^{62.3} \approx 5.76 \times 10^{18}$. Source: Tomás Oliveira e Silva, "Empirical Verification of the 3x+1 and Related Conjectures." In "The Ultimate Challenge: The 3x+1 Problem," (edited by Jeffrey C. Lagarias), pp. 189-207, American Mathematical Society, 2010.

  • The page [6] by Tomás Oliveira e Silva states that, in 2009, they verified the conjecture up to $2^{62.3}$.

  • Earlier, in 2008, Tomás Oliveira e Silva [6] tested all numbers below $19\times 2^{58}$.

  • Much earlier, in 1992, Leavens and Vermeulen verified the convergence for all numbers below $5.6 \times 10^{13} \approx 2^{45.67}$. Source: Leavens, G. T. and Vermeulen, M. "3x+1 Search Programs." Comput. Math. Appl. 24, 79-99, 1992.

  • By the way, the paper [7] from 2019 confirms to me that the largest integer being (consecutively) verified is about $2^{60}$, referring to above sources.


When I put it all together, I get the upper bound $2^{66.4}$.

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