Cycle members mod cycle length

[mathkook]

Just for fun, surely known, and possibly posted … consider a (putative) qn+1 cycle of length l, and consider its cycle members mod l.

3n+1, l=2
cycle: 1, 2
mod l: 1, 0

5n+1, l=5
cycle: 1, 3, 8, 4, 2
mod l: 1, 3, 3, 4, 2

5n+1, l=7
cycle: 13, 33, 83, 208, 104, 52, 26
mod l: 6, 5, 6, 5, 6, 3, 5

5n+1, l=7
cycle: 17, 43, 108, 54, 27, 68, 34
mod l: 3, 1, 3, 5, 6, 5, 6

3n-1, l=1
cycle: 1
mod l: 0

3n-1, l=3
cycle: 5, 7, 10
mod l: 2, 1, 1

3n-1, l=11
cycle: 17, 25, 37, 55, 82, 41, 61, 91, 136, 68, 34
mod l: 6, 3, 4, 0, 5, 8, 6, 3, 4, 2, 1

181n+1, l=15
cycle: 27, 2444, 1222, 611, 55296, 27648, 13824, 6912, 3456, 1728, 864, 432, 216, 108, 54
mod l: 12, 14, 7, 11, 6, 3, 9, 12, 6, 3, 9, 12, 6, 3, 9

181n+1, l=15
cycle: 35, 3168, 1584, 792, 396, 198, 99, 8960, 4480, 2240, 1120, 560, 280, 140, 70
mod l: 5, 3, 9, 12, 6, 3, 9, 5, 10, 5, 10, 5, 10, 5, 10

The 181n+1 cases are kinda structured-looking, though not entirely predictable.

Because 181 \equiv 1 (mod 15), we wind up with a simple residue sequence rule:

r \rightarrow 8r (mod l), for even n
r \rightarrow 8r + 8 (mod l), for odd n

So if n \equiv 5 mod 15, then its successor is either 10 mod 15, or 3 mod 15. Hence the repetitious-looking mod l sequences.

[StaceyStarwolf]

Hi Mathkook,

I wanted to show you a special mod. One where you replace all numbers by the sum of its integers. For example number 11265 will be 1+1+2+6+5 = 15 and 15 will be 1+5 = 6 turning 11285 into mod 6. It also works for negative formulas and their loops, like for example:

Formula -3n+1, cycle 13:
cycle: 13, -19, 29, -43, 65, -97, 146, 73, -109, 164, 82, 41, -61, 92, 46, 23, -68, -17, 26
mod: 4, -1, 2, -7, 2, -7, 2, 1, -1, 2, 1, 5, -7, 2, 1, 5, -5, -8, 8

Formula 3n+1, cycle -17:
cycle: -17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34
mod: -8, -7, -1, -1, -1, -5, -7, -1, -1, -5, -7

Formula 181n+1, cycle 27:
cycle: 27, 2444, 1222, 611, 55296, 27648, 13824, 6912, 3456, 1728, 864, 432, 216, 108, 54
mod: 9, 5, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

I found out that the /2 connections will loop 5, 7, 8, 4, 2, 1 and for some formulas also loop 3, 6, 3, 6 and 9, 9. Same thing for the negative side. The positive connections vary, so I created this legend for them:

This made it possible to create schematics of all the relationships between the negative and positive connections. There are 18 schematics in total which will repeat itself for both positive and negative formulas. For example 19n+1 will have the same schematic as 1n+1 and, -73n+1 will have the same schematic as -5n+1 and so on.

Interesting is that for some formulas the negative loops are reversed. for example the 3n+1 and 15n+1 formulas have the same schematic, but reversed negative loops.

I doubt it will be of any use besides being an awesome poster for your office. But I hope you like it anyways!

Edit: I fixed a few incorrect links.