Interesting patterns in Collatz Conjecture

In this video, I attempt to show an approach to finding a couple interesting patterns for the infamous Collatz Conjecture, aka "3x+1 problem". I also provide formulas and a few graphs that clearly indicate these patterns which do not seem to have exceptions. I have scanned the currently developed patterns and graphs on the Internet and decided to share.

[quantum]

I know it might feel bad reading all the comments saying that this has been done before, but you should feel good. This video is what math is all about: Using your own logic and capabilities to teach yourself (and others) about a topic and perhaps discover something new. Bonus points because you are doing it to pursue your own interest and because you clearly enjoy it. Great video and keep at it!

[anonymousanonymous]

This has been tried before (as practically all amateur approaches to collatz). It is not too difficult to generate a formula for all the different types of numbers that decrease below themselves in n steps as a function of n. What is difficult is showing that the set of all these numbers spans (or fills) the set of natural numbers. I can give such a formula for some simple cases inductively. Consider a number n=(4a-1)/3, where n is a natural number. We can see this implies a congruent 1 mod 3. This gives every number with 3 steps until it is smaller than itself. We can do the same with all number of steps by simply specifying the order of the sequence (div 2, do 3n+1) and then reverse it through modular multiplicative inverse. Again, the difficulty is in showing these sequences cover all the natural numbers.

[HeavyMetalMouse]

For some interesting fun, consider expanding the domain of the problem into all integers: we quickly discover that there are other endpoints, and even loops:
Some things eventually fall into the {-1, -2} loop, suggesting that negative numbers might just fall to -1...
But then we then discover the {-5, -14, -7, -20, -10} loop that never falls to 1
It is trivial to see that no negative number ever 'flips' into the positive realm, and also vice versa

Thus, the conjecture expands a little - there are more than two possibilities: A chain can go to 1 (and get into the {4, 2, 1} loop, or a chain could go off to infinity, or a chain could find itself caught in some other as yet unknown loop.

Ultimately, a proof that every natural number eventually reaches a smaller natural number would, by induction, constitute a proof that all natural number chains go to 1, since no other loops of positive numbers exist within the 'base case' of the induction argument, so there are no other loops to fall into. If a loop exists somewhere in the higher numbers, then there would have to exist at least one natural number greater than 2 that never reaches a number smaller than itself.

I love how tricky and weird this unsolved problem is while still being entire comprehensible and accessible. :)

[evanev]

This is really cool! This is one of the cases where yes someone has done it before, but that's almost everything in maths and that's not a good thing. I think this kind of amateur but well presented maths should be celebrated, even if someone did it in 2006. Treat maths like art!

[vishnudast]

the piano part is new 😄. i have gone through this path as well. I've scripted and plotted graphs in python. but couldn't reach anywhere. Love to see similar minds.
but to be honest, many professors and brilliant minds have worked on this. I'm certain that they must have considered these all. probably with in the first hour after hearing this problem.
But then again i remember this quote all the time. " Sometimes it is the people no one imagines anything of who do the things that no one can imagine. ". So Keep working on it bro. I would really love it, if someone without phds solves it first. 😁

[timurpryadilin]

i understand that this is merely a coincidence that this pattern looks like a piano, but still, this is very very cool ! much cooler than many popular math coincidences. i am genuinely impressed with your ability to recognise a familiar pattern in such an unrelated context. this is the first sign of a great mathematician.

[Turalcar]

The reason it looks like a piano is because in music the 3/2 ratio is also important. 19 is the difference (or sum of differences) over 12 steps. 2^19=524288 is close to 3^12=531441 but not exactly so it eventually diverges.

[konradk]

"piano pattern" stops working when we calculate count of steps for 57115. (200steps mod 31 = 14 unfortunetly )

[thelocalsage]

whenever i find an integer sequence while playing with a math problem, i always scan it through the OEIS database! this one is code A102419, if you wanna go look at some of it. and don’t feel like your contributions to mathematics have to be solving impossibly difficult problems—i love math, and i know i won’t solve the Collatz conjecture. still, sometimes i play around with it and tons of other math problems too tough for me to solve just to scrape at how they work, like walking through a garden and thinking about the mycelial networks beneath and the xylem and phloem of the trees and the pollination symbiosis with the bees and letting it all accent the beauty of the flowers and world even though i’ll never be a true expert in ecology

[binathiessen]

The pattern repeats with powers of 2 works because the next N steps of the collatz conjecture are uniquely determined by the N least significant bits of a number, where the steps of the collatz conjecture are defined as:
x/2 if x is even.
and (3x+1)/2 if x is odd.
Which makes sense because each of the above steps essentially removes the least significant bit of the number.

[philambda]

I did my own experiments using values up to 4.000.000. Cutting at every 12th- step, I got the following "piano"-patterns:

[2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3]
[2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 5]
[3, 2, 3, 3, 2, 3, 2, 3, 3, 12, 8, 21]

(The vaules in the last two patterns are propably incorrect, since they still can change if we calculate more values)

Note you get only 4 "piano"-patterns starting on the note f#. Then it changes to 4 "piano" -patterns starting on the note c#. Then it propably changes to a g# scale. So both times it went up 7 halfnotes (a perfect fith). Don't think this pattern continues forever thogh, as my mathematical instincts that the periodicity would be linked to log_2(3) being rational, which it is not. It'd be funny if the collatz-sereies generated a full cricle of fiths before the pattern breaks, though :D

[BeauBreedlove]

The piano patten isn't cyclical. It breaks at 138. Still I really enjoyed the video.

[MayurGarg]

Interesting analysis. I also tried dealing with this and the Goldbach conjecture. My approach typically involves simple logic that translates the problem into another that is a corollary of the first. So in essence, you prove one, you prove another. Here are the rather simple ones I found myself after a few hours of thought (which I do believe others may have come up with too but I too found them without any inspiration):
Collatz: Instead of proving every number goes to 1, you can prove every odd number goes to a number of type 4^k where k is a natural number. Proving one proves the other.

Goldbach: Instead of proving that every even composite number is a sum of two primes, you can prove that every number greater than 1 is equidistant from at least one set of 2 primes. For the latter, the primes themselves have trivial solution with a distance of 0. For example, 3 is equidistant from 3 and 3 at a distant of 0. For 4, it is (3, 5) at distance of 1. For 10, it is (7, 13) at distance of 3.

[Radnugget]

This was a fun watch and keep up trying to do things like this, even if people already discovered this path, it just proves that it is correct and that even without outside knowledge it can be done.

[DR-]

remember: there is no better discovery in math than rediscovery, for rediscovery must mean you took steps in the right direction

[taproot]

It makes sense that 2^n would show up in multiple places when looking at this function.

Because an entailment of the Collatz conjecture is that, if you the function for long enough, it will always hit a power of 2 in a finite amount of steps.

[columbusmyhw]

Fun fact: that repeating pattern of 12 is also the pattern of the lengths of months in a year (30 or shorter (Feb.) -> 2, 31 -> 3)

[aurigo_tech]

Interesting! Despite every sequence ending at 1, or it shooting off to infinity (which itself would need a proof if such a case would be found) there could also be the possibility of having a loop, like 1-4-2, but with a lot more and huge numbers.

[dalek]

There is a simple formula for the calculation of distances. Basically, only those numbers are interesting which increase at least after the first sequence step. One can write these numbers in the form 4x+3.

For example, if we start with the number 135, the sequence results:
135 (2) 203 (2) 305 (2) 458 (2) 229 (2) 344 (2) 172 (2) 86

We multiply the denominators and get:
2 * 2 * 2 * 2 * 2 * 2 * 2 = 128

We subtract 128 from 135 and get 7 with the sequence:
7 (2) 11 (2) 17 (2) 26 (2) 13 (2) 20 (2) 10 (2) 5
2 * 2 * 2 * 2 * 2 * 2 * 2 = 128

The general formula:
div(u) = div(u-2^n) = div(u+2^n) = 2^n
with div(u-2^n) > 0

If we take u, we get an infinite number of distances with powers of 2:
div(u) = div(u+i*2^n)

In the interval [u, u+2^n] there can be further equal div(u), so that we finally get a sequence pattern with distances of 2^n, that repeats infinitly.

Since there are infinitely many different div(u), this approach is not suitable to prove the Collatz conjecture.

[TheHappyWhale]

I feel like the Collatz Conjecture would be fun for people, even those who don't like math, if it could be gamified in a fun way. Might be something I look into in the future.