LELegionMammal978·Mar 30, 2023
Looking at the alleged proof on your GitHub page (https://github.com/dankearney/collatz/blob/2dd21babd5fbced73...):
> As the cycles get longer, the nearest power of 2 actually trends further away. Why? https://mathoverflow.net/a/116960 explains this nicely -- powers of 2 and 3 cannot be close to one another as powers get larger. We need a power of 2 to be close to a power of 3, but that is not possible.
You misinterpreted the MathOverflow answer. Quoting its result, "At least half of the powers of 2, in a density sense, have a distance from the powers of three of at least one fifth of their size." Notice that it does not set any bound that applies to all powers of 2. Therefore, your assertion remains unproven.
For an explicit counterexample, let's take your n = 1000000001. Let e = 10439860591 and o = 6586818670. Then, 2^e ≈ 3^o + 2.796·10
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^3142711177, but (3 + 1/n)^o ≈ 3^o + 2.206·10^3142711189. Therefore, 3^o ≤ 2^e ≤ (3 + 1/n)^o, contrary to the strong form of your assertion.
Also, you have not proven the nonexistence of sequences which diverge to infinity.