Basically, 3x-1 gives a concrete example that this ms represents no progress whatsoever on 3x+1. Just like all of the references it cites.

]]>There could be a way to prove conjecture:

1) An impair number will always be followed by a pair.

2) A pair number will either be followed by an impair and then see point 1, either by a pair number.

3) No number is repeated within one series.

4) If the number is a power of 2, the series will converge to 1.

5) from Points 1,2 and 3 we see that unrepeated pair numbers will be generated infinitely, except if the series reaches a number which is a power of 2 (point 4).

6) The probability of generating a power of 2, when generating infinite number of pair numbers is 100%, so it is 100% sure that the serie will reach a power of 2 and then converge to 1.

Best Regards,

Sandor ]]>

In either case there is a formulation of the main result without slow-growing functions, as “[lim sup] log-density of exceptions (Collatz orbit stays above C) goes to 0 as C increases” but it wasn’t immediately obvious which sense of lim-sup applies.

]]>I don’t have any plans to work on this particular refinement of the results (and all my current graduate students and postdocs are busy with their own projects and/or are working in rather different areas of mathematics than the ones that are relevant here), so you (or anyone else) are certainly welcome to pursue it if you wish.

]]>Maybe I extend too much, but I want to take the opportunity to ask if Professor Tao’s strategy would change if instead of using mod 3 ^ n, would use mod N, where N is the starting point, to see if zero residue mod N is avoided.

Another question I wanted to ask is whether it is interesting or possible to try to prove that if we assume that no N falls into a periodic orbit, it could be shown that no N escapes to infinity, and therefore the truth of the conjecture would depend solely on demonstrating that no it is possible to fall into a periodic orbit. Thanks. ]]>