Collatz Journal #3 - Mountains of Madness
December 23, 2025•389 words
We're already learning a lot about this problem we're up against but if we're going to embark on this journey we need to get a map of the territory.
First, there's the mountain range of any given Collatz sequence like 27, as we've seen:
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Its parts are:
S1 - the start. If we're interested in only rising sequences then S1 must be odd (2X+1)
Peak - the highest number reached by a finite Collatz sequence. By definition this must be an even number, otherwise its successor would be higher
Drop - this is the first number in the sequence that drops below the start S1
To prove the Collatz conjecture, we could just show that every sequence, every Collatz trajectory, follows this pattern and MUST have a Peak then eventually hits its Drop point then inevitably falls to 1.
The other important map of the territory is the number line itself.
1, 2, 3, 4, 5, 6, 7, 8, 9, ... U1 ... to infinity
If we want to DISPROVE Collatz we need to find (or at least prove it exists) the first/lowest number N that rises or loops forever. Let's call this elusive unicorn U1. While there may be a infinite herd of unicorns (U2, U3, U4, ...) higher than U1, we only need to find one to disprove the mighty conjecture.
Well, it's not all that hard to write a program that runs numbers through the Collatz sequence from, say, 1 to a million and see if we find the unicorn U1. As you can imagine, this has been done and every single number from 1 to million (no matter how high its sequence rises) eventually succumbs to gravity and falls back to 1. In fact, people have run sequences up to 268 (that's a 100-digit number!) and not found any number that defies the Collatz conjecture.
Alas, mathematicians are real sticklers and demonstrating that a million numbers (or a trillion-trillion numbers) all agree with the Collatz conjecture does not constitute PROOF.
At least now that we have a map of the territory (the number line) and a picture of what the kind of mountains Collatz sequences create, we can start to think about where this elusive unicorn might be hiding and where it cannot.
Next:
- Properties of numbers
- Eliminating hiding places for unicorns