3x + 1 Problem. Syracuse Conjecture

Abstract

In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same pattern: 1) Upward trajectory. Odd and even numbers alternate until the cycle reaches an upper bound 2) Downward trajectory. Two or more consecutive even numbers follow until it reaches another odd number. At this point, it's the beginning of the following cycle. Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle. Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.12. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.