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Well, studying a conjecture for "almost all" numbers doesn't actually prove anything about the conjecture because just a single exception would disprove it. The
by Grue3 7y ago
Well, studying a conjecture for "almost all" numbers doesn't actually prove anything about the conjecture because just a single exception would disprove it. There are a few conjectures that had particularly large exceptions [1]. It's possible that Collatz exception will be so large that it's effectively uncomputable or it's existence is unprovable in ZFC [2].
Somebody probably tried it, but I wonder if describing particularly long sequences where r_(n+1) = 3*r_n+1 is always odd would help. There are results for arbitrarily large arithmetic progressions having some specific property [3], so there's no reason why other progressions couldn't be studied in that way.
[1] https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture
[2] https://en.wikipedia.org/wiki/Zero_sharp https://en.wikipedia.org/wiki/Zero_sharp
[3] https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
- mkl 7y agoThe rule is that if r_n is odd, r_{n+1} = 3 * r_n+1, so because of that condition, 3 * r_n+1 will always be even.
- deleted 7y ago[deleted]
- Grue3 7y agoYeah, I forgot that little fact. It should be (3*r_n + 1) / 2 then.
- bluGill 7y agoThat depends on how well we can come up with the "almost all" that fits. If almost all just leaves just 10 numbers not covered it is easy to test those 10 and now you have proved it by exhaustion. If almost all is still an infinite set though that method would fail. It still can be useful though because now we know to focus on that small that don't fit into the previous proof which might lead someplace. Or it might not - this is a difficult problem.
- JoeAltmaier 7y agoAlmost all would be a good result, especially if the exceptions could be identified and characterized. That'd just change the conjecture e.g. "All numbers except those with more than 37 prime factors above 18 quadrillion" or some such.