Talk:Collatz Conjecture/Supposed Proof
In the set of even numbers 50% are divisible by 2 but not 4. 25% are divisible by 4, not 8 25% are divisible by 8, 16, ...
I don't understand why this need explanation?
- Because it is not guaranteed that even and odd numbers appear equally frequently in a Collatz sequence. Some numbers appear more than others, because the sequences themselves overlap. Hence it does not just follow. --prime mover (talk) 22:11, 15 January 2021 (UTC)
Moved from the body of the proof, where it does not belong
This a statement of fact!
- No it is not. It is an observation based on looking at a grand total of $3$ cases!
- "Oh look, 3 is prime. 5 is prime. 7 is prime. Therefore all odd integers are prime!"
There is only one case where the CS increases. That is when the 3x+1 term is 0mod2. The CS will increase on the average iff the average % of the cases in which this happens is greater than 66%.
- SO PROVE THAT THE AVERAGE IS WHAT YOU SAY IT IS. YOU HAVE NOT DONE THIS./ THERE IS NOTHNING OF ANY VALUE WHATSOEVER IN WHAT YOU HAVE POSTED.
The stuff I wrote in the beginning was supposed to show that the growth/loss rates were constant and based on the divisibility of the 3x+1 terms.
- But it didn't. You looked at 3 cases and extrapolated from there.
All of the various rates have a '3' in the numerator and a '2' in the denominator. This strongly suggests that there should be no loops and that every "even number" will eventually become a term in an infinite Collatz Sequence.
- "Strongly suggests" does not "Prove".
The logical flow is not apparent and is further confused by change of variable, e.g. "Let N = 4k+1 Then 3*n + 1 = 12*k + 4", and the lack of defining concepts, e.g. the gain/loss based on the divisibility of the 3n+1 term appears to be very different from the gain or loss shown in the table (which is simply the "Ratio" raised to the power of "No. of terms").
I suspect the aim is to demonstrate some form of limit which tends to zero and at best this would only show some form of convergence. Ultimately, to prove/disprove this conjecture, it needs to be demonstrated that cycles are [im]possible and I do not see where the existence of cycles is proved/disproved.
This appears to be a variation on a theme of the non-rigorous heuristic argument mentioned in https://terrytao.files.wordpress.com/2020/02/collatz.pdf