Abstract
We analyse some probabilistic methods for the proof of the conjecture and we provide a comparison of the proofs with another one found in the literature, using the Reliability Integral Theory and the SPQR Principle.
After, we show a new proof (probabilistic) by merging the infinite number of states into three SuperStates: the merged process is still a Markov process easily solvable.
Later we provide a different probabilistic proof of the Conjecture via the Reliability Integral Theory and the SPQR Principle. We devise a “ideal machine” (Gedanken Experiment) which, as quickly as one wants, makes transitions between the SuperStates and finally ends into the “Collatz Cycle” where it stays forever.
Eventually we show a non-probabilistic proof using Flow Graphs. This last method, based on the flow-graph theory, provides the means to solve completely the problem.
