A permutation of the positive integers defined modulo 3: 3-adic dynamics and cycles of period at most 5000

In a notebook entry dated 1 July 1932, Lothar Collatz considered a function defined by cases according to the remainder modulo 3, which gives rise to a permutation of the positive integers, and asked whether the orbit containing the number 8 is finite or infinite. The question remains open. This note shows that the function is a permutation, gives its explicit inverse and proves that every orbit is either a finite cycle or an infinite orbit tending to infinity in both directions. It proves that the extension of the function to the 3-adic integers is equivalent to the full shift on three symbols, and deduces an exact characterization of cycles by their sequences of remainders, from which there are only finitely many cycles of each period. It also gives congruences for the smallest and largest elements of a cycle, bounds for the proportion of terms divisible by 3, an explicit bound for the smallest element and a relation with the continued fraction of the base-2 logarithm of 3. Combining these results with an exact computer search, it proves that the only cycles of period at most 5000 are the four known ones. The record contains the English version and the original Spanish version.