First, I'm going to use an implicit even step after the odd step, as 3*odd + 1 always equals even. If you look at the path a number takes to its next lowest number, for example 5->8->4, visualize it by just looking at the even and odd steps like so: 5->10, you will see that other numbers follow a similar pattern:
9->10
13->10
17->10
What do these number have in common? They follow the pattern 5 + k(2^n) where n is the number of even steps (with the implicit even step, two in this case).
For another example, look at 7:
7->1110100
Seven even steps, so the next number will be 7 + 2^7 = 135:
135->1110100
I'd love to hear if this has been found and documented somewhere. If not, I have additional ramblings to share.
Of course, if you can identify which n satisfies the equation x = s + k*2^n for some value of n and some "base" value s (7 is the base value in the previous example), you can predict the path of that number.
As an example, take 7 + 4*2*7 = 519. Its next lowest number is 329. 329 = 5 + 81*2^2. So for 329, s=5, k=81, n=2. So we know 329 will only take two steps to reach 247.
Pick a power of 3 n to be the coefficient for k on the right/reduced side, and then the left side will have at least one valid reducing form with coefficient power of 2 f(n) = ⌊n·log2(3)⌋+1. If there is more than one, they will have different constants. Each multiplication immediately has a division (you already got this part), and there must be a final division which is not immediately preceded by a multiplication because (3x + 1)/2 > x for all positive integers (that is, if you multiply once and then divide once, you will always be larger than just before those two things, so an "extra" division is needed to reduce). This means that there must always be at least one less multiplication than division, so the initial condition is one division and zero multiplications - the even case with n = 0. Then for n = 1 you need 2 divisions, which works because 2^2 > 3^1. Then for n = 2 you need 4 divisions, because 2^3 < 3^2 so 3 divisions is not enough. This is where f(n) comes in, to give you the next power of 2 to use/division count for a given n. When you do skip a power of 2, where f(n) jumps, you get an "extra" division, so at 16k + 3 -> 9k + 2 you are no longer "locked in" to only the one form, because there is now an "extra" division which could occur at any point in the sequence...
Except it can't, because you can't begin a reducing sequence with the complete form of a prior reducing sequence, or else it would "already reduce" before you finish operating on it, and it so happens that there's only one non-repeating option at n=2.
At n = 0, you just get D (division). At n = 1, you have an unsplittable M (multiply) D pair MD and an extra D. The extra D has to go at the end, so your only option is MDD. At n = 2, you appear to have three options for arranging your MD MD D and D: DMDMDD, MDDMDD, and MDMDDD. But DMDMDD starts with D so isn't valid, and MDDMDD starts with MDD so also isn't valid, leaving just MDMDDD.
At n = 3 there are finally 2 valid forms, 32k + 11 -> 27k + 10 and 32k + 23 -> 27k + 20, and you can trace the MD patterns yourself if you like by following from the k = 0 case.
The constants don't even actually matter to the approach. If there are enough 2^x k - > 3^y k forms when n goes off to infinity, which it sure looks like there are though I never proved my infinite sum converged, you have density 1 (which isn't enough to prove all numbers reduce) and this angle can't do any better.
Where does the 10 come from?
8 ((5*3+1)/2) is even, so that's where the 0 comes from
4 (8/2) is the end.
Mind you, it's almost certain Collatz is true (generating these classes out to 3^20 nets you just over 99% coverage, and by 3^255 you get 99.9999999%) but this approach doesn't work to PROVE it.
Anyhow I wrote a single-threaded collatz "benchmark" that does this using bigint and its hilarious to run it up around 127 bit numbers, inlet it run for 3 or 4 days and it never finished the first number it was given.
My github has a Java and Python version that should produce identical output. Collatz-gene or so.
edit: I'm in way over my head. Disregard me :)
1. It does happen. These conjectures can fall apart after seeming like a lock: https://en.m.wikipedia.org/wiki/Mertens_conjecture
2. Even if it is true, the process of proving can yield interesting insights.
keepamovin•4mo ago