CollatzIntro  Some general remarks

Workshop 

This
collection of articles is not meant as a general overview on the problem. There
are some better articles available in the net, see for instance of
[Lagarias],[Schorer],[Conrow]. I focus some
special topics, prominently the loopproblem, which I even split up into
smaller portions, to disprove simpler questions first, like the nonexistence
of a "primitve loop" (which are identical to that
"1cycles" or "mcycles" of [Steiner] and [Simons/deWeger]
except for the notation of the parameters). The
conjecture can be proven resp disproven in few aspects: CC: (CollatzConjecture) to show that all
numbers transform to 1 after a finite number of transformations T() : T(n;A,B,C,...Z) = 1 CC can
conversely be proven by an inversed approach: p1) all
numbers can be constructed by the inverse transformation C(), starting at 1 CC can be
disproven if d1) there
exists a number which is a member of a divergent trajectory; where the values
under transformations T() grow infinitely OR d2) there
exist a group of numbers (different from 1) where the transformations T() go
into a loop, thus starting at a number n, doing a transformation
a'=T(a;A,B,C,D,...) the result a' equals a OR d3) the
inverse transformation C() cannot create all numbers if started by 1: CC can be
proven if p2) no
divergent trajectory exists AND p3) no loop
exists. Although I
have invested plenty of time in p1 and found some interesting arguments, this
is currently not my primary focus. Instead I mainly deal with d2) resp p3),
the disproof of a loop. 
Also in the
attempt to *dis*prove I setup some inequalities, assuming some exponents in a
transformationformula like b = T(a;A,B,C,D) to show, that
"a" never can equal "b"  without referring to a special
value of "a". This simple remark may be needed for some Collatzfans,
since in such cases some available articles and hommages to the problem
discuss it from the view of focusing actual elements of a transformation,
instead of focusing the characteristics of a certain type of transformation,
which requires a certain possible or impossible structure for its elements a
and b. So I won't
enter some "sportive" approaches: which is the highest number that
has such and such a trajectory, since with the notation of b =
C(a;A,B,C,D,...) one can
construct many types of examples just by respecting some not too hard modular
requirements for the sequence of exponents A,B,C... 
From the
definition of my T()transformation it is obvious, that I only deal with odd
numbers as elements of this transformation. For all questions, that I want to
discuss this is no restriction, but even a simplification. 
For instance,
the inverse transformation C() provides a handy tool to generate a very
instructive tree, which I have hoped is such simple, that it could prove the
CC by p1. [sheet01] The study of
this tree led to many interesting graphs, either as spreadsheet of as
graphical picture (my most favorite is a fractal tree in the form of a
bottlebrush with whiskers and infinite selfsimilar repetitions) [fractalgraph]. The sheet01, displayed in
base4numbersystem exhibits a much simplifying structure [sheetbase4],
which is not evaluated finally. 
The notation b =
T(a;A,B,C...H) gives raise
to display some general formulae. 
A transform a'=T(a;A,B,C) can be
written as a' =
((((a*3+1)/2^A)*3+1)/2^B)*3+1)/2^C) which can be
expanded to a*3^3
+ 3^2 + 3^1*2^A + 2^(A+B) or more
general, where N indicates the "length" of the
multisteptransformation and S may indicate the sum of all exponents a*3^N
+ 3^(N1) + 3^(N2)*2^A +
3^(N3)2^(A+B) +...+ 3*2^(A+...+F) + 2^(A+...+G) or the canonical
form: 3^N 3^(N1) + 3^(N2)*2^A
+ 3^(N3)2^(A+B) +...+ 3*2^(A+...+F) + 2^(A+...+G) From that
canonical form it can also be easily derived that 3^N and thus an "a"
can be found easily, if the "standardized" transform T(0;...) is
computed and then which can be
useful to find a pair of integral a> a' which satisfy the structure of
the specified transformation. 
The canonical
form exhibits some interesting results: ·
For
any combination of exponents we can find one "a", which can be
transformed to a valid "a'" ·
There
are infinitely many solutions for finding "a", namely all the same
residue class base 2^S ·
The
higher the sum of exponents s, the higher must the value of "a" be
to result in a valid "a'" 
Conversely,
if we study the inverse transformation a =
C(a',H,G,...C,B,A) we find 2^S 3^(N1) + 3^(N2)*2^A
+ 3^(N3)2^(A+B) +...+ 3*2^(A+...+F) + 2^(A+...+G) and the
Collatzconjecture CC implies this way, that each odd number can be created
by a series of fractions 2^S 1 2^A 2^(A+B) 2^(A+...+F) 2^(A+...+G) or with
S'=SN, A'=A1>=0, B'=B1>=0 ... d = 3/2 2^S'
2^(A'+...G')
2^(A'+...F') 2^A' 2^1
with
appropriate exponents A,B,C,...G,H 
which I find a remarkable result in itself. 
Finally, this
canonical form leads on a simple path to the formula, which describes the
loop. In that case the lhs and rhs must be equal, thus with a'=a
thus forming a loop a(2^S 
3^N) = T(0;...)* 2^S T(0;...)* 2^S and for a
loopcandiate of the form
3^(N1) + 3^(N2)*2^A + 3^(N3)2^(A+B) +...+ 3*2^(A+...+F) +
2^(A+...+G) where the rhs
must result in an odd integer>1 This formula
also occurs, if the problem is attacked from an system of linear equations
involving equations for all intermediate transforms a,b,c,...g,h 

last update: 15.8.2004 


