Some papers associate Collatz Conjecture to Prime Numbers properties. Others consider it a Number Theory problem. Since my mathematics don’t go that far, I started to see Collatz Conjecture as a battle between even and odd numbers. That battle didn’t make much sense to me, until I noticed the “hidden rules” of the sequence:
The “classic” definition
can be expanded to
This way, the conjecture turns to be a battle between odd and “power of 2” numbers.
One advantage of this approach is that this 2 groups are evenly distributed on the new Collatz sequence: after a odd number there is a “power of 2” number, and so on. Whether the sequence tends to infinity or 1 depends of the randomness distribution of “power of 2” group at the sequence and its effective “weight” – by what factor is n divided.
We can be sure that the odd factor is 3 – each odd number is multiplied by 3.
About “power of 2” numbers, we know their distribution. Each “power of 2” 2
x is 2
-x of all even numbers.
The sum converges to 1.
The division factor for each 2
x are given by the rules : it’s 2
x.
The “weight” of each “power of 2” is given by its factor
distribution
The total weight for all “power of 2” numbers (which is the factor we are looking for) is given by the product of the series
For all “power of 2”, the maximum factor is 4, so we calculate the ratio odd/”power of 2” as 3/4.
Considering that all odds are multiplied by 3 and all “power of 2” are (statistically) divided by 4, Collatz function can’t diverge to infinity because of the Law of large numbers. Does it always reach number 1? Maybe there is a repeating cycle that excludes 1, different from 4→2→1 (which in the expanded Collatz function is reduced to the cycle 4→1), but bear in mind that at range {1..4} the “power of 2” factor is ≈ 2.83, so the ratio odd/”power of 2” is close to 1. That never happens at other ranges.
It can be assumed that at Collatz function variations like 5n+1 most of the numbers will diverge to infinity, because of the 5/4 ratio. Why not all? If a sequence hits a “power of 2” like 1→6→3→16 then you’re stuck in a cycle. How there is a cycle with such a divergent ratio? You can find similar cycles for 7n+1 and bigger ratios. I again call for the Law of large numbers to affirm that this happens solely with small samples, where the “power of 2” distribution is not respected by the sequence. The bigger the sequence, the most improbable to find such cycle. The bigger the numbers, the more sparse are the infamous “power of 2”.
Since you can find some 5n+1 sequences that reach 1, I guess the next step is to find out if 5n+1 sequences always hit a power of 2. If true, that function will also never diverge to infinity.
