Observations towards a proof of the Collatz conjecture using $2^{j}k+x$ number series

Abstract

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller number, all full sequences lead to 1 with a finite stopping time, the problem could be reduced to more structured shorter sequences. It is shown that this sequences exist and follow consistent rules. Potential features of an infinite cycle, also leading to a smaller number, are also discussed. Further, an argument for only one possible closed cycle is given for the special sequence of alternating odd and even steps as well as arguments that infinite cycles must exist. Using the observation that periodic behavior is exists an additional argument is provided the probability of subsets which will end up at a number smaller the initial value possibly even of N will indeed end up at unity in the Collatz problem [1]. The work is to be seen as in progress and shared as an contribution to discussion rather than a concrete publication.