The structure of the 3x + 1 problem

Alf Kimms (Chair of Logistics and Operations Research, Mercator School of Management, University of Duisburg–Essen, Lotharstr. 65, 47048 Duisburg, Germany)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications

Paul Erdös said about the 3x+1 problem, "Mathematics is not yet ready for such problems". And he is seemingly right. Although we cannot solve this problem either, we provide some results about its structure. The so-called Collatz graph is iteratively transformed into a sequence of graphs by making use of some hidden structure information. It turns out that the transformation of graphs corresponds to a sequence of sets of numbers. It is shown that if the union of these number sets were equal to the set of integers greater than one, the famous Collatz conjecture would be true.