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I know none has been found, and there probably isn't one considering the effort people have put into it, but has it been proven? (for some reasonable definition of "closed form"). I'm mostly interested in proving the nonexistence of closed forms for sequences starting at a given $n$.

Elliot Gorokhovsky
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  • Do you mean a closed form for the stopping time (for the hailstone sequence from a given starting value $n$ to reach $1$)? – Joffan Jun 10 '16 at 18:26
  • Either a closed form for hailstone_sequence_starting_at_n(i) or stopping_time(n) – Elliot Gorokhovsky Jun 10 '16 at 18:27
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    Well, it hasn't even been proved that the stopping time exists for all $n$.... – Greg Martin Jun 10 '16 at 18:36
  • @GregMartin That's irrelevant. Proving no closed form exists is a weaker result that doesn't depend on finiteness. Indeed, if the stopping time doesn't exist, that would just settle the question. Conversely, if it does exist, the question remains open. In no case does it matter one way or the other. – Elliot Gorokhovsky Jun 10 '16 at 23:50
  • I sketched a proof of the Collatz conjecture a couple of years back. There may be no closed form per se but there is a set of closed forms of increasing resolution, each of which provides all solutions up to some n, and in the limit as the "resolution" of the closed form increases, the number of integers captured approaches infinity. The proof appears to be closely related to the p-adics. I'm pretty sure I can prove the conjecture with the help of a good mathematician; Invite me to a chat if you want to work together on it. There's about a week's work in it. – Robert Frost Jun 14 '16 at 13:21
  • There is a continuous version of Collatz here http://math.stackexchange.com/questions/30536/continuous-collatz-conjecture and I think you will have more luck converting that into a closed form than the discrete version. – Robert Frost Jun 16 '16 at 07:42

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