Collatz-Intro - The loop-problem |
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The loop-problem is the question, whether there exists any transformation with finite many exponents A,B,C where the output-number a' equals the input-number a: That means for a transformation a'= T(a;A,B,C,...H) that a' equals a, where a is an odd number>1. The latter restriction is made, since there is the "trivial" loop 1 = T(1;2,2,2,2...2) for any number N of exponents "2" In the following I may omit the clause "... except the trivial one..." for readability. |
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It is very simple to prove, that there cannot be another loop of length 1 except the trivial. This is shown here to introduce the reader into my concept and the main formula, how to disprove the possibility of a loop in general without much things around. Also it is possible to exclude certain loops up to lengthes around 100..200 steps (which means about 200...300 steps of the original definition of the collatz-transformation) by an elementary proof using obvious bounds for the exponents. Some results, however, are already known in the collatz-research. These are all results from using approximation-tools, and say things like: since we know, all numbers below 2^40 are known not to belong to a loop, a loop - if it exists must have the minimum length of 250000 steps or the like. The interested reader may visit the [Rosendaal]-site for up-to-date results. Since I were unable to contribute to that research, but more, because my interest is more an analytical proof, I never entered this path of search. Even more, it seemed to be a promising attack, to use a sophisticated system of modular classes, which had already helped me in understanding the rules of transforms. This means to eventually disprove the possibility of such a looping with arguments of modular arithmetics. This was backed from the canonical formula for a loop, since a loop is disproven for such cases, where nominator and denominator cannot not be divided without remainder. But unfortunately this problem could not be solved yet despite some good progress in the beginning, and was therefore put away awaiting a later consideration. Since the general loop (with arbitray exponents) could not be successfully be attacked this way so far, I decided, first to study a simple form of a loop, such that it is ascending several steps, and then descends with one single additional step. I called this type of loop "primitive loop" and got very far with a disproof. Recently I found an article, that just this "primitive-loop" was successfully attacked by Ray Steiner, who used the name 1-cycle for that construct, and was able to disprove the possibility of such a loop with results of theory of rational approximation with transcendent numbers. But since that result requires some deep insight in this theory, I continued my try to find a more elementary proof. There exist disproofs for "general loops" up to the length of 250000, as I stated above, disproofs for "primitive-loops" of arbitrary lengthes, and even for loops consisting of concatenated primitive-transformations, called "m-cycle" up to the number of 68 concatenates (which includes, that each involved partial primitive transformation is of arbitrary length) |
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It is also worth to be noted, that all that approaches to the disproof must somehow be aware of deep connections between the involved parameters 3/2/1 of the collatz-transformation, since the analoguous stated 5x+1-problem indeed has loops, and even short ones, and also primitive loops. Keeping the equivalent formulae of that 5x+1-problem in mind sometimes helped to prevent to try too general formulae and to return to the specifics of the 3/2/1-relation. Especially in the question of the "primitive loop" (1-cycle) very direct connections to some unsolved general questions related to the ratio of logarithms of 2 and 3 are immediately visible, so it is in any case worth to watchout the proceedings in that area of research. |
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last update: 15.8.2004 |
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