Collatz-Intro - Notations

Workshop
Recreational
Mathematics

Gottfried Helms Univ. Kassel

mailto: helms at uni-kassel

www: math-homepage

 

 

 

I found some sources, which use an analoguous definition for the notation of the collatz-transformation, but I think, that my proposal here is worth to be favored.

The original Collatz-problem seem to have been stated in the following form:

 

For any natural number>0 define a Collatz-transformation from a to b

 

CT_odd:= b = 3*a +1 if a is odd

CT_even:= b = a/2     if a is even

 

That means, after a CT_odd always a CT_even-transformation must follow, but it is possible, that many CT_even-transformations follow each other.

The first enhancement in notation could then be to put all CT_even together in a parametrized transformation

 

CT_even1:=     b = a/2^A  with A being the exponent, which makes b an odd natural.

 

 

I put both transformation together in one formula and call that transformation

 

T(a;A):=  b = (3a+1)/2^A

 

In this notation "A" normally is a parameter, which is not free but exactly determined by the value of "a" (this might confuse the appropriate understanding of the underlying problem).

But for a general analysis it is useful,

 

·         if we deal with analytical manipulations

·         if we deal with a generalized view of the problem, where we allow intermediate rational values for a

·         if "a" is unknown beforehand and the collatz-process is analyzed regarding such exponents

·         if we deal with the inversion of the operation.

 

The iteration of such transformations ("multi-step-transformation") may be noted as

 

T(a;A,B):= T(T(a;A);B)

 

to any extent, example

 

b = T(a;1,4,2,1)  means

b = ((((a*3+1)/2)*3+1)/16)*3+1)/4)*3+1)/2

 

 

The inverse operation means

 

C(b;A):=  a = (b*2^A-1)/3

 

where A is now a free parameter. C(b;A) is the inverse of T(a;A) and

 

T(C(a;A);A) = a

 

Note that the inversion of a multi-step-transformation requires to write the exponents in reverse order:

 

b = T(a;A,B,C)

a = C(b;C,B,A)

 

For the sake of not wasting too many letters in longer formulas, I sometimes write this

 

a' = T(a;A,B,C,...)

 

The quote is currently not meant as a derivation-symbol or a transposition in all these following text.

 

 

 

 


                                                                                                                                last update: 15.8.2004