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Consider the following recursive formula: $$x_{n+1}=x_n-\frac{1}{x_{n}}$$ with $x_0=2$. Is the sequence produced by this formula unbounded?

Some time ago I came across this problem somewhere on the internet. According to this paper on the subject, the question was open at the time it has been published. But I can't seem to find any evidence wether it has been proven already or wether it is still open. Does anybody know?

Of course a proof is more than welcome if you happen to know how to solve it.

EDIT: I don't understand why my question is being marked as a duplicate. It clearly asks something else than the suggested duplicate, namely wether the problem has yet been solved or not, while the duplicate asks about the problem itself. Also, the redirects on the suggested duplicate, and redirects on the redirects, don't seem to provide an answer to my question either. Am I missing something here? [No longer relevant, however I'll leave this text for clarity.]

EDIT 2: This is the page my question was (falsely) marked as a duplicate to and this is another interesting question about the same problem. I thought I'd add these pages, because they contain some ideas regarding the question and since I get the feeling it is still unanswered, it might be nice to have these accessible from one place.

Tyron
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    @Koro Why do you think the sequence is decreasing? Try evaluating it something like 10 times, and you will see it jumps back and forth in a chaotic fashion. – Tyron Apr 02 '20 at 09:35
  • Right! My bad! It isn't a decreasing sequence. – Koro Apr 02 '20 at 09:54
  • It decreases until $|x|<1$ then blows up and then decreases again gradually until $|x|<1$ again. Since the amount it blows up to depends on a pseudo-random remaining fractional value after a long series of decrements by small amounts, it seems very likely to be unbounded. – Suzu Hirose Aug 14 '22 at 07:32

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Not an answer but too long for a comment.

The most likely answer is no. ResearchGate doesn't have any known citations on that paper, as well as all the references being used dating to before 2000, so before Graham posed the question. Marc Chamberland has only $4$ papers on the arXiv this one not among those $4$ and Mario Martelli has none. Apparently this paper was indeed published according to ResearchGate (which had many more Marc Chamberland papers). I could not find any other papers talking about this problem

My guess would be is that this is not an interesting enough question. Meaning that I'm guessing the reason Graham posed it was not because that dynamical system had any relevance, but to highlight our lack of understanding of dynamical systems, to the point where even questions as simple as that one are out of our reach at the moment. I'm guessing a solution to that problem would be way deeper than just what the actual problem is, developing much more our understanding of those dynamical systems, but I might be wrong. I think the problem was probably just a placeholder for any problem that seems like we should know the answer to, but we don't

Bruno Andrades
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