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
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
inverse transformation C() cannot create all numbers if started by 1:
CC can be proven if
p2) no divergent trajectory exists
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.
b = T(a;A,B,C...H)
gives raise to display some general formulae.
can be written as
a' = ((((a*3+1)/2^A)*3+1)/2^B)*3+1)/2^C)
which can be expanded to
+ 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
+ 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
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)
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)
with S'=S-N, A'=A-1>=0, B'=B-1>=0 ... d = 3/2
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
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) +
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