TL;DR
This paper investigates the divergent and convergent behaviors of trajectories in the 3x+1 and 5x+1 problems, revealing that a significant portion of integers diverge in the 5x+1 case while most seem to converge in the 3x+1 case.
Contribution
It provides the first proof of divergent trajectories in the 5x+1 problem and analyzes the distribution of trajectories in both problems, offering new insights into their long-term behaviors.
Findings
Over 17% of positive integers diverge in the 5x+1 problem.
The percentage of divergent trajectories in the 3x+1 problem tends to zero.
Values ending in 1 tend to grow larger over time.
Abstract
We will prove that there are trajectories generated by the function at the origin of the 5x+1 problem which are divergent. The iterative application of this function on the set of positive integers allows us to determine that more than 17% of all these integers start divergent trajectories. Regarding the 3x+1 problem, this percentage tends towards zero, suggesting that all positive integers are part of converging trajectories. Despite this appearance, we cannot conclude that all positive integers belong to convergent trajectories. Nevertheless, the results obtained in this paper allow us to follow the evolution of the distribution of trajectories and to understand why the values of positive integers ending in the integer 1 are more and more large.
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Taxonomy
TopicsBenford’s Law and Fraud Detection · Digital Media Forensic Detection · Computability, Logic, AI Algorithms