It is difficult to say whether someone has looked into the Collatz Conjecture through the approach I will be demonstrating. From what I can tell, this may be a first.
The basic principle of our conundrum is multiplying an odd number by 3 and adding 1, this by default creates an even number which will then be divided by 2 until it becomes an odd number again. This process is repeated again and again and it is difficult to prove that all numbers will eventually reach down to the number 1, although this is all one can observe through trial.
I will hereby endeavor to try and shed some light on some of the inner workings of this conjecture.
One initial aspect is to notice similarities within the patterns generated which leads to the following observation: all numbers can be categorized by their number in the ones and by being preceded by an odd or even number. That is, for example the number 1199 has 9 in the ones and is preceded by 119 which is an odd number.
This has quite an important role to play in deciphering the inner workings of the numbers that follow or precede any given number. If we put our number through the steps initially described, we get the following sequence: 1199 → 3598 → 1799 → 5398 → 2699 → 8098 → 4049 → 12148 …
We can see a pattern emerge oscillating between 8 and 9 in the ones. This pattern is no coincidence and is one of 3 that lead to a continuous increase by multiplication intertwined with a single division.
The basic principle behind this pattern is that any number with 9 in the ones preceded by an odd number will lead to a number with 8 in the ones, preceded by either an even or odd number. If the 8 in the ones is preceded by an odd number, it will in turn divide only once into a different number with 9 in the ones, and if preceded by an odd number, the cycles repeats itself. This repetition is not random and can be predicted. In short, o8 (odd number preceding an 8 in the ones) will lead to an o9 (odd number preceding a 9 in the ones) creating an ascending cycle.
The two other cycles that lead to an oscillating increase are o4→e7→e2→o1 and e6→e3→o0→o5 in the ones. For example: 1246 → 623 → 1870 → 935 → 2806 → 1403 → 4210 and 2174 → 1087 → 3262 → 1631 → 4894 → 2447 → 7342.
After analyzing every single number in the ones preceded by odd or even numbers, a map is created:
One thing that becomes clear is that multiples of 2 have quite an important role to play. We can write all odd numbers in one or another form of a multiple of 2 by subtracting 1.
For example, 3 can turn to 2² − 1, 7 to 2³ − 1, 11 to 2² x 3 − 1, etc.
Subsequently, if we apply our formula, for example, to the number 7 we get the following:
3 x 7 + 1 = 3 x (2³ − 1) + 1 = 3 x 2³ − 2 = 2 (3 x 2²) − 1
Once we go through the second step of division by 2, we end up with 3 x 2² − 1. If we repeat this process we, will get: 3² x 2 − 1 and 3³ − 1.
Through this we can see that the representation of any number in powers of 2 gives critical information about the number of upwards oscillations before we reach a drop.
2ⁿa − 1 will always lead to 3ⁿa − 1 where n is the number of oscillations before reaching a dropping point and where 3ⁿa − 1 is a multiple of 2.
In our above example we get 7 → 22 → 11 → 34 → 17 → 52 → 26 where 26 is a dropping point towards 13. The drop itself follows a pattern and is predictable.
1199 = 2⁴ x 3 x 5² − 1 As we can see, we can expect 4 oscillations reaching 3⁴ x 3 x 5² − 1 before a drop:
1199 → 3598 → 1799 → 5398 → 2699 → 8098 → 4049 → 12148 → 6074 → 3037
We can now see why certain numbers might have surprising yet not so unpredictable trajectories.
We can also see that multiples of 2s and 3s follow the same trajectories.
If we generalize this approach, we get a neat representation of all numbers and eventually some patterns emerge.
Even numbers can be organized in powers of 2 multiplied by 1, 3, 5, 7 and 9 as stepping anywhere within a column will result in a straight cascade down, ex: 928 → 464 → 232 → 116 → 58
Odd numbers can be organized in powers of 2 multiplied by 5 and a series of 1, 3, 9, 7 subtracting 1.
These tables provide vital information about the inner functioning of the conjecture since 2ⁿa − 1 will step by step lead to 3ⁿa − 1, we can track certain patterns of movement within the tables as after each oscillation the power of 2 drops by 1 and the number multiplied by triples.
For example 39 will lead to 59 and then 89: 2³ x 5 − 1 → 2² x 15 − 1 → 2 x 45 − 1
These patterns are most beautifully illustrated in the table of 9s as tripling the multiplication factors of each column leads perfectly to a different column, though they reproduce perfectly well within the whole system.
To map the drop after each oscillation we have to figure out how many times the number in question can be divided by 2.
If we take a rather convoluted approach for the sake of demonstration, the number 19 can be written as 2² x 5 − 1, which through a 2-step ascending oscillation will lead to 3² x 5 − 1. This can be written as (2²−1)² x (2² + 1) − 1 = 2⁶ − 2⁵ + 2⁴ − 2³ + 2², and as we can see, through factorization we will have 2², that is a 2-step drop:
19 → 58 → 29 → 88 → 44 → 22 → 11
This can be mapped out in the following way:
Two patterns emerge in parallel every 2 odd numbers, one alternating between a 1-step and a 2-step drop as above and the second following a pattern with internal growth: 1 – 3 – 1 – 4 – 1 – 3 – 1 – 5 – 1 – 3 – 1 – 4 – 1 – 3 – 1 – 6 …
This eventually grows into larger and larger drops.
Within the table above, at predictable intervals, a “hiccup” occurs as highlighted where the pattern of a line repeats itself (numbers 25 and 79).
Within the table below, all numbers have been put at descending intervals for the sake of visualization, with the misfits highlighted and the correct number of drops in parenthesis.
If we take as an example the same number 1199 = 2⁴ x 3 x 5² − 1, we can predict a 4 step increase by oscillation to 3⁵ x 5² − 1, followed by a single step drop.
Or if we take a more evident example, 23 = 2³ x 3−1, we can predict a 3-step oscillation to 3⁴ − 1, followed by a 4 step drop.
This, for example, can help us map out numbers that will drop below their initial value after the initial upwards oscillation as per the table below.
With all of the information discussed so far, we can even map out the movements of all trajectories and their sub-branches such as in the table below where each color represents a given trajectory.
Below are two examples of numbers that drop into the same trajectories: the numbers 5119 and 133119, and the numbers 1919 and 10239.
If we take it the other way around, starting from 1 and going up, we can trace all numbers starting by some main branches that split into secondary branches and so on. The trajectories are predictable and have a predictable flow.
One way to think about it would be of hair with split ends that wrap around a pyramidal screw. The split ends will in turn split themselves creating intricate patterns that follow a clear rule of ascent, or descent, but no loop can ever form nor will a number forever meander without dropping to 1 as we can categorize all numbers within the table, following the very same patterns and the very same trajectories as the ones demonstrated as examples.