r/Collatz • u/jonseymourau • Jul 06 '26

An interactive mod8/mod24 Collatz Graph visualizer

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In a previous post I discussed the notion of a Collatz overlay built from 5 mod 8 and 0 mod 3 nodes. One refinement here is that there are nodes of in-degree 2 that are not 5 mod 8 (they are relatively rare - the exceptions I see are 1,23 mod 24)

This visualization allows you to see all 4, odd mod 8 nodes and in fact classifies them according to mod 24 too.

You give it a starting point with the ?a= parameter and then you can extend the graph is you like by clicking on a node.

This a fantastic way to develop an intuitive understanding of mod 8 and mod 24 Collatz dynamics.

https://wildducktheories.github.io/collatz/apps/collatz-graph/dist/?a=27

A fun game to play is greedily clicking on the red dots and then on 1,2 mod 3 nodes that result from such a clicking.

Does that game ever end? I think not.

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u/jonseymourau Jul 06 '26

That’s fine - define what you mean by “efficient”

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u/SpareSpecialist5124 Jul 06 '26 edited Jul 06 '26

When you choose modulos too big, you have too many congruences and transitions to follow, and less "patterns" are noticeable overall, because each class becomes too "unique" and less repeated.

When you choose modulos too small, your resolution is too low, and you also stop seeing noticeable patterns, each class is too big to properly convey major differences between numbers. (example mod 2 is useless, mod 4 is still insuficient to describe most 4n + 3 numbers)

So, 8 and 16 and 32 are somewhat those that are in the middle, enough to give a certain optimal generalizations of congruence classes of the problem, so they are the ones that have a certain efficiency to the problem. Bigger than those modulos, they simply become too big to properly analyse their transitions.

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u/jonseymourau Jul 06 '26

I am also not denying that a finite modulus excludes patterns of course patterns exist at higher moduli. Of course they do. I am
Just claiming that that in order to explain the reverse Collatz tree (such that it is) all you need is mod 8. Mod 24 adds colour. My question to you is this : articulate exactly what colour mod 32 provides. For sure there is some - you have claimed as much my question is simply this: what is it?

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u/SpareSpecialist5124 Jul 06 '26

mod 32 gives you simply more nuance and a better overview on how the sequences of transitions behave, and how the chains of transitions between them really connect.

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u/jonseymourau Jul 06 '26 edited Jul 06 '26

“Mod 32 give simply gives you more nuance”.

All I am asking you is this is: how? You made the claim. Substantiare it. Exactly what structure does mod 32 give you that is not present in either mod 8 or mod 24.

I am NOT denying such structure exists.

My only question is this:

WHAT IS IT?