Wikipedia : Collatz Conjecture

Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach $1$.

To quote:

Steiner (1977) proved that there is no 1-cycle other than the trivial (1;2). Simons (2004) used Steiner's method to prove that there is no 2-cycle. Simons & de Weger (2003) extended this proof up to 68-cycles: there is no k-cycle up to k = 68. Beyond 68, this method gives upper bounds for the elements in such a cycle: for example, if there is a 75-cycle, then at least one element of the cycle is less than $2385\times 2^{50}$.

Has there been any legitimate progress since then, in terms of cycles or anything else?

  • $\begingroup$ @ParclyTaxel If you are sure, then should we close this question? $\endgroup$ – Jaideep Khare May 13 '17 at 15:57
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    $\begingroup$ Jeffrey Lagarias has two annotated bibliographies on the Collatz conjecture, for those who are too lazy to search for this research themselves, like the OP. They can be found on the arXiv, and describe hundreds of research papers on this conjecture. $\endgroup$ – TMM May 13 '17 at 16:06
  • $\begingroup$ @Parcly I assume you're being sarcastic. $\endgroup$ – TMM May 13 '17 at 16:09
  • $\begingroup$ @TMM I know, right? But I sincerely do think that while this conjecture is interesting, it's less important than (say) Riemann or lonely runner. $\endgroup$ – Parcly Taxel May 13 '17 at 16:10
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    $\begingroup$ "Maybe you should do the research yourself." I was on the fence about leaving this open (in review), but you've dissuaded me. $\endgroup$ – hardmath May 13 '17 at 22:33

Fields Medalist Tao has a blog post on The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 from August 25, 2011.

Quoting Tao:

"Needless to say, I didn’t solve the problem, but I have a better appreciation of why the conjecture is (a) plausible, and (b) unlikely be proven by current technology, and I thought I would share what I had found out here on this blog."

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  • $\begingroup$ Interesting, but I still would like to see further developments. $\endgroup$ – Yuval Levental May 13 '17 at 17:32

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