Are There Structural Symmetries Between Collatz-Type Loops? (5n+1 Examples)

General
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I’ve been looking into Collatz loops under the 5n+1 variation, and I noticed something when talking with @cheshirecatalyst and comparing two cycles:

       13 → 66 → 33 → 166 →  83 → 416 → 208 → 104 → 52 → 26 → 13  
Steps:     u    d   u     d    .u.    d    .d.    d    d    d
Steps:     u    d   u     d    .d.    d    .u.    d    d    d
       17 → 86 → 43 → 216 → 108 →  54 →  27 → 136 → 68 → 34 → 17

The step patterns, up and down, are identical except for a “swap” between where those two are in the necklace.

This got me thinking: when a loop exists in some Collatz, is there some structural or algebraic criterion that might require some other loop with the same length also exist? It reminded me of how in Riemann hypothesis, any off-line zero would by symmetry also imply 3 others (complex conjugates and algebraic symmtetry of equation).

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Not sure if that answers your questions, but some “collatz” are similar to others in that they are guaranteed to contain the same structures (reduced 3n+1 and 3n-1 have respectively the cycles {[0],[1,2], [-1], [-5,-7,-10], [-17,-25,-37,-55,-82,-41,-61,-91,-136,-68,-34]} and {[0], [-1,-2], [1], [5,7,10], [17,25,37,55,82,41,61,91,136,68,34]}, that share the same structure).
Some other are guaranteed to contain all the structures of another one plus some new ones. For example 3n+3 contains the cycles [0], [3,6], [-3]… but also new ones like [1,4,2] (udd) that doesn’t exist in 3n+1.

I have a more complete analysis of those patterns for “collatz problems” that goes 1n+k and I am confident something interesting could be found about the 3n+k, the 5n+k, etc.

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It’s great to study the existing loops to understand why they happen. When would the existence of one loop imply the existence of another? It’s great.

For 5n+1, the most promising loop-shapes are ones with length k=7 and total number of odds x = 3, because loop members have the form n/(2^7-5^3) = n/3. There are only five loops like that:

1110000 39/3 = 13-33-83-208-104-52-26-13
1101000 43/3 = non-integer loop, starting at 43/3
1011000 53/3 = non-integer loop
1100100 51/3 = 17-43-108-54-27-68-34-17
1010100 61/3 = non-integer loop

(1 = “up/odd”, 0 = “down/even”)

You can calculate whether a particular parity sequence leads to an integer loop – it would be interesting to be able to say “if parity sequence X corresponds to an integer loop, then so does parity sequence Y”. (Or, “if parity sequence X doesn’t… then parity sequence Y doesn’t.”)

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For the proposition, “If the 5n+1 loop with parity sequence 1110000 consists of integers, then the 5n+1 loop with parity sequence 1100100 also consists of integers” … there actually is a reason for it, a way to show it.

1110000 39/3 = 13-33-83-208-104-52-26-13
1100100 51/3 = 17-43-108-54-27-68-34-17

The formula for the first member of the 1110000 loop is a = \cfrac{2^0 5^2 \cdot 2^1 5^1 \cdot 2^2 5^0}{d}, where d = 2^7 - 5^3 = 3.

Likewise, the first member of the 1100100 loop is b = \cfrac{2^0 5^2 \cdot 2^1 5^1 \cdot 2^4 5^0}{d}.

Because b and a share a lot of terms, their difference is simple:

b-a = \cfrac{2^4 5^0 - 2^2 5^0}{3} = \cfrac{12}{3} = 4.

So, b = a+4.

That means if a is an integer, then so is b, and in fact, a=13 and b=17.

This reasoning is nicely based on the parity sequences, but I feel it’s not too generalizable … the “swap” of 100 for 001 at the end causes an increase of 12 in the numerator, which happens to be a multiple of 2^7 - 5^3 = 3.

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I’m interested enough to take a crack at it in lean. It “feels” like there’s an argument or more general principle here.

Some observations/potential connections:

Binding d, one may choose up to 2^d-1 for the “parity” or “rotations of the up-downs”.

(in 3n+1… ) My mental model for these has been sums of 3-smooth numbers A003586 - OEIS or A348175 - OEIS

It seems trivial to show that terms like this (in 3n+1) are able to be summed to every value in the range, I have not given too much time to 5n and higher on this.

For 3x+1 see an old demo of mine: Game

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The generalization seems to check out, some Fermat’s little theorem and other standard modular tricks. It seems to me the more challenging question being determining when/where/if this applies to generalized collatz in some way that hasn’t been done before.

For any m*x + 1 Collatz-like mapping (where m is odd).

for loop denominator d = |2^k - m^x| (where k = total cycle length, x = number of odd steps).

Two parity cycles of length k differ by swapping a 1 at index i and a 0 at index j (and share the same power of m in their numerator positions).

If the distance between swapped indices i and j is a multiple of the multiplicative order of 2 under d then one of those being and integer is necessary and sufficient for the other being an integer.

Another way gemini walked me thru this:

d divides 2^p - 1 (Assumed order)

2^p - 1 divides 2^(i-j) - 1 (Polynomial identity)

Therefore, d divides 2^(i-j) - 1 (Transitivity)

Therefore, d divides 2^j * (2^(i-j) - 1) (Multiplication)

Which is exactly 2^i - 2^j .