Collatz-Intro - Some general remarks |
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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 disprove 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 may be unusual for many Collatz-fans since in most cases the available articles and hommages to the problem discuss it from the view on the elements of a transformation, instead of the characteristics of a certain type of transformation, which require 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 structur 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, ithis 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 |
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last update: 15.8.2004 |
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