A cool organic shape created from a graph of the collatz conjecture
The collatz conjecture is a very famous mathamatical statment, named after Mathamatician Lothar Collatz. It essentially states the means of a simple function with two rules. If you take any positive natrual number and if it is even divide it by two and if its odd you multiply it by three and add one, it always will result in a loop of 4, 2, 1. Now initally this seems somewhat questionable but I can assure you it is very much true. Take any number say... 23. Now we apply the function that we previously discussed. (and feel free to try this yourslef. I can assure you you will not find any exceptions!) Lets see what happens when we apply these rules.
Sample Seed: 23
Ok, so we first apply the rule to the seed. its odd so 3 * 23 + 1 = 70
We continue to put the result through the function...
70 / 2 = 45
45 * 3 + 1 = 136
136 / 2 = 68
68 / 2 = 34
34 / 2 = 17
17 x 3 = 51 + 1 = 52
52 / 2 = 26
26 / 2 = 13
(13 * 3) + 1 = 40
40 / 2 = 20
20 / 2 = 10
And here is just about where we enter the loop
10 / 2 = 5
5 * 3 + 1 = 16
16 / 2 = 8
8 / 2 = 4 / 2 = 2 / 2 = 1
1 * 3 + 1 = 4; 4 / 2 = 2 / 2 = 1
And there we reach the loop!
Just attempt it with any number. I can assure you the result will be the same
A few interesting finds have been discovered, and its been found that over short distances we experance some degree of randomness with the values generated in the function. But it seems that the pattern trends downwards over large distances explaining this innate trend towards 4, 2, and 1. This can be pointed out by the following graph.
I also have come across a few observations maybe explaining this phenomena and why it trends downward, always hitting 1; that perhaps at least partially regards the structure of numbers in math in a simper way. Mathamaticlly we can define even or odd numbers as these equations where n is a number of pairs
Even: 2n
Odd: 2n + 1
And if we think about it it makes sense. Even numbers can be split into pairs. Odd numbers cant be evenly divisable, hence why they are even plus one. We can use this to partially explain the collatz conjecture. All even numbers are divisable by two. When an odd number is multiplied by three and one is added we recive an even number. Although this is not sufficent enough to fully explain this pheneomena, perhaps for whatever reason natrully this results in an inevetiable downward trend into 1 which results in the loop.
Thus far devoted mathamatitions have tested over 2^68 numbers and as you probably thought- the same loop has been yeilded. Its belived that some extremly large, but finate number for whatever reason could possibly disobey this law for whatever reason either through the form of a seperate loop or one that instead of trending downward as we discused trended upwards. Although 2^68 seems like a lot of numbers, we could techniclly continue infinatly, and to but it in perspective another famous conjecture the Polya Conjecture was eventually proved false at a number much larger than the collatz conjecture. Never the less this problem still proves its self fastninating.
Perhaps eventually a number will be found that disproves the conjecture. Should we even bother to search? Paul Erdos, a well known mathamatician once said. "Mathamatics is not yet ripe enough for such questions". Maybe one day we will have an explanation for exactly why this happens too
Well to conclude the collatz conjecture is a very intresting occurence. If you would like to try some numbers out just to see for yourself, I have made some very simple programs in C, Python, and C++ that will run the function on an inputed seed, that you can feel free to use!
