I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or proof?

Thank you in advance.

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    $\begingroup$ I think the only similarity is that no one has any idea how to settle either of them. $\endgroup$ – Gerry Myerson Sep 18 '17 at 6:19
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    $\begingroup$ More specifically, they're both particular instances of problems whose generalisations are $\Pi^0_2$-complete. $\endgroup$ – Adam P. Goucher Sep 18 '17 at 7:44
  • $\begingroup$ @AdamP.Goucher good, but possibly there are much deep link. $\endgroup$ – XL _at_China Sep 18 '17 at 8:39
  • $\begingroup$ The continued fraction for $\log 3/\log 2$ comes up in Sinisalo, On the cycle length of the Collatz sequence, www.probleme-syracuse.fr/cargo/Sinisalo_collatz.pdf but that's not what you want if for no other reason than that that number isn't algebraic. See also math.stackexchange.com/questions/172476/… and mathematics.pitt.edu/sites/default/files/… and d-scholarship.pitt.edu/24817/1/Masters_-_Collatz.pdf and darrellcox.website for more of the same. $\endgroup$ – Gerry Myerson Sep 18 '17 at 13:13
  • $\begingroup$ Both problems are questions about the dynamics of certain transformations, though very different ones: Collatz conjecture is about the behavior of natural numbers under the Collatz map (discrete situation), while continued fractions are basically analysis of the map $\frac{1}{x-\lfloor x\rfloor}$ (continuous operation) on the set of algebraic numbers (which is somehow "discrete algebraically", but indiscrete topologically). Is this perhaps what you meant? $\endgroup$ – Wojowu Dec 21 '17 at 9:52

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