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I'm thinking of a problem in number theory in which one applies a recurrence that's something like doubling $n$ if it's even and taking it over $3$ if it's odd...but with some ceilings or additions added in. The conjecture is that every number's orbit eventually gets down to $1$, and I believe this is unproven. I think there's about a 60% chance I encountered this problem first in Godel, Escher, Bach by Douglas Hofstadter, where there might also be some quote about this problem being far beyond the resources of contemporary mathematics. It also seems like Hofstadter calls numbers that eventually come down to $1$ "marvelous," "miraculous," or some such, but Googling such terms got my nowhere.

Does anybody have any idea what problem I'm talking about?

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In the book you qoute it appears in a dialog between the turtle and Achiles, after which Achiles betraid the turtle and hand him over to the cops. – Eran Aug 9 '13 at 11:05

You're talking about the Collatz conjecture, a.k.a. the $3x+1$ problem.

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That would be the Collatz conjecture/hailstone sequence.

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