Collatz-Intro - The primitive loop

Workshop
Recreational
Mathematics

Gottfried Helms Univ. Kassel

mailto: helms at uni-kassel

www: math-homepage

 

 

PL: Introduction

Introdruction to the "primitive loop"

The term "primitive loop" is a private notion, resulting from the problem, that the analysis of the general loop was too complicated to derive decisive results. To keep the formulas simpler for a start, I decided to focus the simple form: some ascending steps- only one descending step.

 

I separate things in four parts:

 

1)    Notations

 

2)    Basic formulae,

2.1)  modular requirements for input and output member of a strictly ascending sequence of transformations

2.2)  modular reqirements for the least element in a primitve loop

2.3)  the critical diophantine equation for the primitve loop

 

3)    a modular view

 

4)    Approximation-arguments

4.1)  the combined critical diophantine equation and approximation inequality

 

 

1)Notations

 

In my notation:

 

primitive Transformation

 

 a' = T(a;1,1,1,1,...,1,A)

 with length N,

      N-1 ascending steps (all exponents are 1),

      1 descending step (exponent>1).

 

 

In newer writing I make use of the further abbreviation

 

    a' = PT(a;N:A)

         indicating:

           N-1 steps with exponent 1,

            1  step  with exponent A.

 

 

 

From that : "primitive loop":

 

 a' = T(a;1,1,1,1,1,1...,1,A) = a

 

or

 

 a' =  PT(a;N:A) = a

 

 

The PT-N-loop

The next step of complexity is then to concatenate several "primitive transformations" into one:

 

 a' = PT( a;N1:A,N2:B,N3:C) 

or

 

 a = PT( a;N1:A,N2:B,N3:C) 

forming a concatenated-primitive-loop (PT-N-Loop)

 

One can see, that this -with lengthes N=1 - can as well describe the general loop: then there is no ascending step between two descending steps, and thus an arbitrary concatenation of primitive transformations allowing N=1 can be seen as a proper description of a general loop.

 

 

 

Note: an analoguous concept was followed by several authors, for instance [Steiner],[de Weger],[Schorer] with the notation "1-cycle", where one step is defined as either

 

 b = (3a+1)/2 if a odd  or

 b = a/2 if a even.

 

Thus the parameters of the "1-cycle" K,L indicating the numbers of ascending resp descending steps relate to my notation:

 

1-cycle(K,L)  equivalent   a = PT(a;K:L+1) = P(a;N:A)

 

The equivalent generalization to concatenation of more of one "1-cycles" (which are no more cycles, btw) is the notation: "m-cycle" in [DeWeger].

 

 

2) Basic formulae

 

To attack the problem for any length of a primitive loop, it is wise, to look deeper into the structure of an "a" which could be the first member of such a longer loop.

 

We know already from the canonical eqution for transformations, that

 

                     3^N     

 a' = T(a;A,B,C) =  ----- a + T(0;A,B,C)     ( with N exponents and S=sum-of-exponents)

                     2^S

 

 

2.1) structure formulae for a primitve transformation for leading and trailing elements

 

Now we let all exponents be 1, and denote the length of this purely ascending steps as L = N-1

 

For such a transformation of the length L also  S equals L.

For convenience of notation let z be the last member of such a repeated transformation then

 

                       3^L      3^L - 2^L

 z = T(a;1,1,1...1) = ---- a + ----------

                       2^L        2^L

 

 

       3^L      3^L - 2^L

 z =  ---- a + ----------

       2^L        2^L

 

 

 z*2^L = 3^L ( a + 1) - 2^L          // multiplying by 2^L

 

 2^L (z+1) = 3^L (a + 1)             // collecting powers of equal base

 

so

 

  z  + 1      a + 1

  ------- =  -------
    3^L        2^L

 

 

thus all a of the structure

 

 

   a = k*2^L - 1                 // with the free parameter k

 

and

   z = k*3^L - 1                  // with the same free parameter k

 

 

satisfy this equation, because

 

  (k*3^L-1) + 1     (k*2^L-1) + 1

  -------------- =  --------------
        3^L              2^L

 

       k*3^L            k*2^L

  -------------- =  --------------
        3^L              2^L

 

        k        =     k          

 

all k satisfy the equation and we from here introduce k as a free parameter.

 

From the general approach, to use odd numbers>1 in a loop (but this restriction can as well been dismissed), we can apply some restrictions to this parameter k, so that it produces valid values for a and z.

 

First we assume always, that all members a,b,c...z of a loop are odd and greater 1,  so from

 

   z = k*3^L -1

 

 

it follows also, that k must be even to keep z odd, thus have the form 2i, thus

 

   z = 2*3^L*i -1      // replacing k by 2*i

 

 

and

 

   a = 2*2^L*i - 1     // replacing k by 2*i

 

 

Now add one decreasing transformation at the end, to make a loop possible

 

  a'= T(z;A)  

 

 

so

 

        z*3+1      (2*3^L*i - 1)*3 + 1
  a' =  ------   = -------------------  
         2^A          2^A

 

         (2*3^L*i - 1)*3 + 1
  a' =  -------------------  
              2^A

 

 

then

 

         2*3^N * i - 2      3^N*i - 1
   a' =  -------------- = ----------
           2^A               2^(A-1)

 

 

finally the structure of a', depending on N and a free parameter i

 

          3^N * i - 1
   a' =  -------------
            2^(A-1)

 

 

That means, for a primitive transformation a' = P(a;N:A) we know the required structure of a and a':

 

A primitive transformation

 

  a' = T(a;1,1,1,1...1,A)  with N steps and L=N-1 ones,

 

  in shorter notation

 

  a' = PT(a;N:A)

 

requires as structures for input a and for output a'

 

  a  = 2^N*i - 1     (from    a = 2*2^L*i -  1  )

 

and

        3^N *i - 1
  a' = ----------- 

        2^(A-1)

 

 with the same free parameter i>0

 

Also we recognize that a', being odd, cannot be divisible by 3 (because the nominator of rhs is not divisible by 3) thus the output of such a transformation can be written as

          +1
a' = 6k {  
          -1

 

2.2) Formulae for the primitive loop

 

To make the primitive transformation a primitive loop, the leading member "a" must match both equations, that for a and that for a', and we have a much more restrictive condition

 

a primitive loop of the form

 

   a = PT(a;N:A)

 

requires for the structure of the first member "a"

 

       3^N *i - 1
  a = -----------            and             a = 2^N *i - 1

        2^(A-1)

 

 

 

2.3) the critical diophantine equation for the primitive loop

 

From this follows the critical diophantine equation for the primitive loop, which must be solvable in integers N,A, and i  (N>A>1, i>0 )

 

 

             3^N*i - 1
   2^(A-1) = ----------
             2^N*i - 1

(where the last condition on i excludes the unwanted case of a=-1).

 

This result suggests two basic modular restrictions:

 

1) i must be odd to keep the nominator even and thus divisible by 2^(A-1)

 

2) a cannot be divisible by 3, but must have the form a = 6i+1 or 6i+5 (see example of one transformation)

 

 

 

 


                                                                                                                                last update: 15.8.2004