Predicted the total number of steps in Collatz conjecture.

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How I predicted the total number of iterations in Collatz conjecture sequence in case of 2^100000-1 and 3^100000-1 with an error less than 0.065%.

In case 2^100000-1 this number in binary is 100000 times 1.
11111111..... (100000 times)

1.When process via Collatz it starts with 100000 steps where the length of the number in binary is extended by 1.58496 bit/iteration to total length 158496 after 100000 first iterations.

2.Starting from this iteration it goes through random process to 1 which shorten the length of this number on average by 0.415037 bit/iteration. It gives additional 381884 steps.

When we sum up 1. and 2. we have total 481884 iterations. In reality it was 481603 steps (0.058% off). This part counted only 3n+1 operations.

To count also div by 2 operations we have:
1.First 100000 it is 1 by 1 so for every single operation 3n+1 is one “div by 2” in total 100000 “div by 2” iterations.
2. In remaining part starting from the length of the binary number 158496 bits, half of this number will have one “div by 2”, 1/4th will have 2 “div by 2” per one 3n+1, 1/8th will have 3 “div by 2” per one 3n+1 and so on. In total this will give 763739 operations “div by 2”.

Summing up 1. and 2. we have 863739 times “div by 2”.

Predicted number of all operations (3n+1 and “div by 2”) is 1345623. In reality it was 1344926 (0.051% off).


In case of 3^100000-1 this number in binary is total chaos of the length 158496. It is easier.
To know number of "odd steps" (3n+1) we know it is shortened on average by 0.415037 bit/iteration. We have 381884 “odd steps”. In reality it was 381603 (0.073% away).
To count number of “div by 2” steps again half of it has one “div by 2” per one “odd step”, 1/4th has 2 “div by 2” steps per one 3n+1, 1/8th has 3 “div by 2” steps and so on. In total it is 763739.
Sum of odd and even steps (3n+1 and “div by 2”) is 1145623. In reality 1144919 (my prediction is only 0.062% off).

My method is explained in the video on youtube “The Proof of Collatz Conjecture - For Dummies”
https://www.youtube.com/watch?v=J8kHjNPsFJY

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Last line should be:
Sum of odd and even steps (3n+1 and “div by 2”) is 1145623. In reality 1144925 (my prediction is only 0.061% off).

[Guest]

Who cares? Stop wasting time...

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Who cares? Stop wasting time...

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[Guest]

Can you do this for any initial number ?