  We will prove that there are trajectories generated by the functions at the
origin of the 3x+1 and 5x+1 problems which are divergent. The iterative
application of these functions on the set of positive integers allows us to
determine that nearly 17% of all these integers, in the 5x+1 problem, belong to
divergent trajectories. Regarding the 3x+1 problem, this percentage becomes
very small, suggesting that all positive integers are part of converging
trajectories. Despite this appearance, we will prove that this percentage will
never be zero and that there will be an infinity of integers belonging to
divergent trajectories, thus leading to the conclusion that the 3x+1 conjecture
is false.
