Is there a known example of a "distant cycle"?

General
1

Let’s consider any problem of the shape “nk+m”. Is there any combination of n and m such that:

  • there are some cycles, b being the higher element of those cycles.
  • all the integers between b and B fall into those cycles, B being an integer that is several orders of magnitude higher than both b, n and m.
  • there is at least one other cycle, with elements bigger than B.

?

Of course, as proving such a thing is impossible would prove Collatz, I don’t expect such a proof, but a counter-example could be nice if it exists.

2

So, looking for a large gap between (a) the highest member of a bunch of cycles and (b) the lowest member of the next cycle up.

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3

I’ve never seen it happen … one place to look might be more general “Collatz-like” rules or “tricots”, where the next-step decision is based on mod d, instead of mod 2 (even/odd). Each of the d branches is of the form n \rightarrow \cfrac{a_i + b_i}{d} when n \equiv i mod d.

Among those kinds of rules, a place to look might be the “barely contracting” or “barely expanding” rules, where \cfrac{a_0 \cdot a_1 \cdot ... \cdot a_{d-1}}{d^d} is close to 1. Those rules have some wacky cycles.

4

I also noticed wacky cycles could happen sometimes, but usually the corresponding tricot also have other wacky cycles and there is no “gap”: wide range of number that all belong to lower cycles before the next cycle starts.

I think that’s where I am heading then. Trying to prove such a gap is impossible. I probably won’t manage to prove it but I have no doubt I will learn something cool on the way :slight_smile:

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5

(n,m) = (3,29)

For every x<2531, there are three possible cycles. The largest element across those three cycles is 392. so b=392

At 2531 we find a new cycle whose minimum value is 3811 and whose maximum value is about 6 million. So B=2531

All of the integers between b and B fall into the three small cycles, but B spawns another cycle with elements bigger than B.

Not quite “orders of magnitude” difference, but the conceptual phenomenon seems to exist.

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