Collatz-Intro - 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 loop-problem, 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
"1-cycles" or "m-cycles" 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
transformation-formula 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 Collatz-fans,
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
bottle-brush with whiskers and infinite selfsimilar repetitions) [fractal-graph]. The sheet01, displayed in
base-4-number-system exhibits a much simplifying structure [sheet-base4],
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
multi-step-transformation and S may indicate the sum of all exponents a*3^N
+ 3^(N-1) + 3^(N-2)*2^A +
3^(N-3)2^(A+B) +...+ 3*2^(A+...+F) + 2^(A+...+G) or the canonical
form: 3^N 3^(N-1) + 3^(N-2)*2^A
+ 3^(N-3)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^(N-1) + 3^(N-2)*2^A
+ 3^(N-3)2^(A+B) +...+ 3*2^(A+...+F) + 2^(A+...+G) and the
Collatz-conjecture 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'=S-N, A'=A-1>=0, B'=B-1>=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
loop-candiate of the form
3^(N-1) + 3^(N-2)*2^A + 3^(N-3)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 |
-
|
|
|
|
