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.


Naming the starting numbers u and the exponent of the digit 2 m the transformation for the uneven numbers can be written as H(u) = (3u + 1)/2m = r with r uneven. The examination with the values m = 1, m = 2, ... leads to the table (TAB1) where the starting numbers are ordered in rows and columns, the m-values are entrees of the rows, the r-values of the columns. TAB1 is complete, free from repetitions, constant and free from cycles: all values appear once and only once; after each transformation operated on all starting numbers, TAB1 is, after rearranging, the same as before; it is impossible, starting from whatever starting number, to obtain the same starting number after whatever number of transformations. We show that successive uptransformations of unlimited length do not exist while they are a much slower rising function than their localisation. Furthermore, since there are no cycles in TAB1, the columns within that table are rearrangable in a totally arbitrary way. This allows us to construct a table of preceding starting numbers which leads us directly to the conjecture of Collatz.

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