Recently, I have been reading about the Collatz Conjecture and on the wikipedia page for it, came across the fact that:

Rigorous bounds
Although it is not known rigorously whether all positive numbers eventually reach one according to the Collatz iteration, it is known that many numbers do so. In particular, Krasikov and Lagarias showed that the number of integers in the interval $[1,x]$ that eventually reach one is at least proportional to $x^{0.84}$.

Would anyone know where I can learn more about this bound (why $0.84$?) or where the paper was published (is it online?). Cheers


closed as off-topic by Did, Dietrich Burde, Michael Grant, user91500, JMP Jan 4 '17 at 11:21

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  • $\begingroup$ I entered Krasikov and Lagarias in Google, and voila (click me)... $\endgroup$ – barak manos Jan 2 '17 at 16:16
  • $\begingroup$ Two clicks search: arxiv.org/abs/math/0205002 Quote: "By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map". $\endgroup$ – Did Jan 2 '17 at 16:16
  • $\begingroup$ The bibliography at the bottom of that Wiki page has the original references. See [19] there. $\endgroup$ – B. Goddard Jan 2 '17 at 16:18
  • $\begingroup$ Why $0.84$? Well, more precisely why $\log_2(\lambda)=0.84175\cdots$ $\endgroup$ – Dietrich Burde Jan 2 '17 at 16:24

The first Google search result for Krasikov and Lagarias is the relevant article on arXiv.


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