According to a translation of the letter of Collatz to Professor Mays in 1980, Collatz mentions that he hasn't figured out whether the number n = 80 resulted in a cycle or not, concerning the collatz function. But this doesn't make sense, since the number 80 is easily computed under the collatz function. I've found the probable copy of the letter in german, but after carefully analyzing it, it isn't clear what is the number, it seems like to be 30 (which would make more sense, but still, its orbit only reaches 160 at most), or 80. My guess is that it should be 27, because this number has a really long orbit almost reaching 10000 and Collatz probably lost his patience to compute it.

The alegated copy of the letter in german can be seen in the link:


And the english translation version can be seen in:


Can someone confirm that 27 is the number, by exhibiting an explicit copy of the letter different from the one that I have? Or do you think the number is either 30 or 80?

I would appreciate any clarifying answer. Thank you.

  • $\begingroup$ I found a proof: arxiv.org/abs/2101.06107 $\endgroup$ Jul 14 at 14:14
  • $\begingroup$ Surely this must be a misprint. If a cycle exists at 80 then so must it exist at 40, 20, 10, 5 etc., and presumably Collatz had determined the orbit of 5 to terminate at 1. $\endgroup$
    – NWR
    Jul 14 at 16:00
  • $\begingroup$ My own translation of the relevant section of the letter: "When I gave number theoretical lectures myself I demonstrated this example and I did the same at conferences and posed it as a problem: Is the number n=80 part of a cycle or not? I myself had only a small desktop calculator available to me, and as far as I could carry out my computations with it, no cycle resulted for n=80 and I wasn't able to answer the question." $\endgroup$
    – njuffa
    Jul 15 at 0:19
  • $\begingroup$ Thank you, but the number 80 doesn't make sense, I guess there should be a misprint in the letter. I'm waiting for someone who could have a legitimate copy of the letter. $\endgroup$
    – Albert
    Jul 15 at 10:14
  • $\begingroup$ @DescheleSchilder While the title of your document says "Complete proof" , the abstract explicitly says it proves the conjecture for specific sets of integers, not all integers. $\endgroup$ Jul 15 at 11:55

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