Collatz Journal #4 - Properties of Numbers

As we've seen, just brute-force trying numbers won't find the Collatz unicorn that returns to one. Neither will it prove that such a number CAN'T exist.

We need to move beyond calculation using specific numbers and think about and imagine numbers we can only name, numbers without a face.

How can that work? How can we talk about numbers (and whole classes of numbers) without naming them exactly? That's the difference between calculation and mathematics.

Let's start by looking at the rules of Collatz: If a number is odd, multiply by 3 and add 1. If it is even, divide by 2.

We have a way to talk about odd and even numbers using algebra which is simply a language to talk about unknown numbers.

An even number is a multiple of 2 which we write as 2X where X is any unknown integer.
An odd number is any number NOT a multiple of 2 which we can write as 2X+1.

To see how this is true, look at a number line:

To fill in the odd numbers in-between, just add one:

So, instead of spelling out this whole table we can just write:

Even = 2X
Odd = 2X+1

Note: We could also write Odd = 2X-1

Armed with these definitions we can rewrite Collatz rules as:

Fall rule: If number is 2X, 2X → X
Rise rule: If number is 2X+1, 2X+1 → 3(2X+1)+1 = 6X+4

Where we use right-arrow (→) to mean "next in Collatz sequence."

Now we're getting somewhere! This exposes two interesting things about rise steps right away:

1. After every rise, we must land on a 6x+4 number. Examples: 3 → 10 = 6x1+4, 11 → 34 = 6*5+4
2. Since 6x+4 must be even (because 6x+4 = 2(3x+2)), then every rise step is followed by a fall step

#2 means we can rewrite the Collatz rules as:

Fall Rule: If number is N = 2X, 2X → X or N/2
Rise Rule: If number is N = 2X+1, 2X+1 → 3(2X+1)+1 = 6X+4 → 3x+2 or (3N+1)/2

Let's call these Successor Rules because they define the Successors of any number N in its Collatz sequence.

As we've seen, since every number in the number line can be expressed as 2X or 2X+1, these rules tell us the Successors of ALL numbers in the Collatz sequence. If you were dropped randomly at any number N in the number line from 1 to infinity, you know you'd land on N = 2X or N = 2X+1 and could predict what would come next in the Collatz sequence. What numbers, though, might come before that random number N? In other words, can we say something about what number might come before N in a Collatz sequence?

Another way to ask this is: What are the possible Predecessors of N that would get to N from a Fall Rule or a Rise Rule?

Be definition, since 2X becomes X every number X could be preceded by higher number 2X.
Now, if we reverse the Rise Rule we could determine whether N could be reached from (N-1)/3.

Let's try that for some numbers:

What is clear is that most numbers (11, 12, 14, 15, 17, 18, 20, 21) can't have an immediate lower predecessor.
because N-1 isn't evenly divisible by 3. However, neither can 13 or 19 because N-1 would be an even number which can't rise according Collatz rule #2. That leaves 10, 16, and 22 which are all 6x+4 and therefore have a lower odd predecessor 2x+1 (3, 5, 7). These calculations nicely confirm what was predicted by the Rise Rule #2 above.

Also apparent is that properties of these numbers repeat when expressed as 6X+b. In fact, we can compress this table and include successors as follows: