This is the website of Raymond Hamers, Ostend, Belgium

Dit is de website van Raymond Hamers, Oostende, BelgiŰ

Interested in the Conjecture of Collatz, I tried to make a proof

Ge´nteresserd in het vermoeden van Collatz, heb ik geprobeerd een bewijs te leveren.



Restricting the starting numbers to odd values, we show that for m=1 the starting numbers are of the form u=3+4t leading to the results r=5+6t, and for m=2 we have u=1+8t leading to the results r=1+6t. These results are obtained by what we call a "splitting off mechanism" presented now as a tree-structure where the starting numbers to be splitted-off form the trunk and the starting numbers being splitted-off and leading to the result r=1+6t and corresponding with even valued m-figures form the left branches and the starting numbers being splitted-off and corresponding with odd valued m-figures form the right branches and leading to the results r=5+6t.

Following, we show that, on basis of that splitting-off tree, a table (TAB1) can be created, constructed in two partial tables TAB1A and TAB1B.

TAB1A corresponds with even m-figures and results r=1+6t while TAB1B corresponds with odd m-figures and results r=5+6t. The starting numbers occupie the fields of this partial tables and some parts are shown in (3.2). The resulting numbers form the heads of the columns and the m-figures form the entries of the rows. We show it is sufficient to know the first resulting number, the first m-figure and the first starting number of each partial table to be able to generated those partial tables until any number of columns and any number of rows we want.
Finally, a colour version of each partial table is created following the code: "1+6t"=yellow, "5+6t"=orange, "3+6t"=green, visualising immediately several properties. The construction of TAB1 creates order in a apparently chaotic world.

Thank to this order we have been able to detect the q-numbers, with q=11+12t being resulting numbers larger than some of their corresponding starting numbers. Those q-numbers obviously are real troublemakers in proving the Collatz Conjecture. And so, we created TAB2 from q to M, M being the first non q-number in transforming the q-numbers. With that table we show that uninterrupted "uptransformations" don't exist in TAB1 and that all M-values are within the series 17+36t, half of them for very special values of t.

Finally, we are able to proof the conjecture, because, on basis of the basic properties of TAB1, we have been able to create a rearrangement of columns in TAB1 leading to a "preceding order of starting trees" allways conducting to the result "1".

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