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@blockspins

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    1. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      5/ probably also where Conway got the idea for his FRACTRAN universal Turing machine https://en.wikipedia.org/wiki/FRACTRAN  So here's the idea: the 3x+1 map is HARDWARE that runs programs, and each initial seed is SOFTWARE, that is, a program that produces some behavior when run on hardware.

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    2. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      6/ This hardware is almost certainly (provably?) not a universal Turing machine, but what does it really do? Every program ever put into it eventually halts at 1, but who's to say there isn't some extremely long program, thousands of bits long, that has some other behavior?

      2 replies 3 retweets 22 likes
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    3. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      7/ From this point of view, you should not at all be impressed with 3x+1 being verified up to 2^60 -- ok, all programs of length 60 characters halt. That says nothing. The heuristic/probabilistic arguments point to average behavior, and are probably right - it's entirely

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    4. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      8/ possible that almost all initial seeds decay to 1, while some extremely special ones don't. Makes the problem really hard, since you don't know what you should be trying to prove! Another close (I think) analogy is to Conway's Game of Life;

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    5. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      9/ it's also a universal Turing machine, because one can find gliders, etc, see https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life … Those gliders were not easy to discover, but they were the key to unlocking the universality of the Game of Life. What if there actually is some 3x+1 software (that is,

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    6. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      10/ some initial seed) that makes something like a "glider" (some kind of understandable repeating but growing pattern) when "run" on the 3x+1 hardware? Let's try to elaborate a bit more. As in [K-Sinai, op cit], it's enough to look at numbers coprime to 6, and divide out

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    7. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      11/ all powers of 2. So the map is x->(3x+1)/2^k, with k>=1 as large as possible, keeping the result integral (notice the images are always coprime to 3). These k's are the key here. The Structure Thm in [K-Sinai] says that, no matter what "k-Path", that is, sequence of these k's

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    8. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      12/ you want to occur in order, say, (k_1,..,k_N), there's an arithmetic progression of x's which are exactly the seeds having the prescribed first N steps. For example, for the k-Path (1,1,2,2), the initial seed =1(mod 6) is 199, which one can check has this sequence. What's the

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    9. Alex Kontorovich‏ @AlexKontorovich 13 Sep 2019
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      13/ point? Here's a pic of a 3x+1 path. Each row is a number in binary; the black dots are 1's and white are 0's. The solid line on the right is that every number is odd. This is the seed 887570260646934643447331259693373543698140232874500619768489156911. You wouldn't know it atpic.twitter.com/qn7hAKeQnm

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    10. ban blue light‏ @robrobutt 15 Sep 2019
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      Replying to @AlexKontorovich

      Would love to see more graphics like this one. Do you have any way of generating them for arbitrary seeds?

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      bgc‏ @blockspins 16 Sep 2019
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      Replying to @robrobutt @AlexKontorovich

      Here's a quick sketch in @observablehq :https://observablehq.com/@bryangingechen/3x-1-paths-in-binary …

      10:11 AM - 16 Sep 2019
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        1. ban blue light‏ @robrobutt 16 Sep 2019
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          Replying to @blockspins @AlexKontorovich @observablehq

          Radical!!

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        2. ban blue light‏ @robrobutt 16 Sep 2019
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          Replying to @blockspins @AlexKontorovich @observablehq

          Ive noticed something interesting after playing around with this tool: pictured here is a neighborhood around ~150th step of the orbit of 2^200 - 1, and then the same image but color-inverted. Notice how structurally similar the two images arepic.twitter.com/buZ94m34p9

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        3. ban blue light‏ @robrobutt 16 Sep 2019
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          Replying to @robrobutt @blockspins and

          I wonder if this points to some dynamically-equivalent bevahiors the collatz map has on N and bitflip(N) for "typically" distributed N

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