Re: [Collatz] was : Re: Status of Waring-problem - Collatz - Sorry

From: Gottfried Helms (helms_at_uni-kassel.de)
Date: 08/28/04 

Date: Sat, 28 Aug 2004 10:14:31 +0200

Am 28.08.04 06:48 schrieb Mensanator:
>>
>> 3^N - 2^N
>> a = -------------- (condition for a 1-cycle or "primitve loop")
>> 2^S - 3^N
>>
>>The solution is only valid if a is odd and is integer. For N=1 and A=2 we get
>>the trivial loop of one step:
>> 1
>> a = C(a;2) = --- = 1
>> 1
>>
>
>
> Does this only apply to positive integers? The reason I ask is my formulations
> span both positive and negative domains and I get two 1-cycle loops:
>
> 1 -> 4 -> 2 -> 1
>
> and
>
> -1 -> -2 -> -1
>
Well, in my notation this would be

                     2^1
    x = C(x;1) = x* --- + C(0;1)
                     3^1

                    3^1 3^1 - 2^1 3^1
    x = C(0; 1) * ---------- = ------------* -------- = -1
                   2^1 - 3^1 3^1 2^1 - 3^1

>From the notation there is no specific restriction on numbers x.
You even can use rationals, if you do some more finetuned analysis.
All these are just compacted notations, no new inventions.

----------------------------

For instance, another thing that you immediately see -if you use this notation-
is, that infinetly many solutions in x' and x exist for a certain transformation-
structure, say

      x' = C(x; A,B,C)

This is
               2^(A+B+C)
     x' = x * ---------- + C(0;A,B,C)
               3^3

The C(0;...)- part is in general a fraction with the denominator 3^N, in this
case of a three-step-transformation it is 3^3.
So you see, that for all x with the same modular-class based on 3^3 you
find an integral solution in x and x' for the transformation x->x' , or
can try to find approriate exponents for a given pair (x,x'), for instance
x' = 2x+1 or x'-x = 2^B or anything the like.

Extended to the question of cycles it is simply demonstrable, that
for a certain sequence of exponents only one solution for x is possible;
that means, a search for possible loops over reduces to a search for
configurations of exponents, and strong restrictions can be stated for
such a sequence: the sum S of all exponents must between about 1.5*N and
2*N and the like.

-----------------------------

Using a certain sequence of exponents A,B,C,

      x' = C(x; 2,2,1,2,1,...)
you can construct numbers x->x' where x has an arbitray long trajectory ,
where the intermediate values never fall below x (I think, this is called
"glide" ?).
There are arbitrarily many transformations x' = C(x; A,B,C..), but to
be applied to a pair of integral (x,x') it seems to me, that this sequence
allows the smallest numbers (empirically and only few tests, maybe there
are easy counterexamples except the trivial one).

--------------------------------

and so on.
As I said, I suggest it as a practical notation for the whole collatz-calculus.

Gottfried Helms

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