The present article analyzes the Collatz conjecture in two fields: loops and divergence. First, the problem of whether there are more loops than the well-known 4 - 2 - 1 cycle is addressed. Through an analysis of the loop formula, it is demonstrated that there are no additional loops. Secondly, the problem of divergence—sequences that tend to infinity—is examined by applying the inverse rules of the Collatz conjecture. The analysis indicates that no such divergence is possible. Consequently, by ruling out both alternative cycles and divergent paths, this work provides evidence that the Collatz conjecture is true for all positive integers.
Carlos Alfonso Martínez Mayor (Mon,) studied this question.