Another Generalization of $3\mathrm{x}+1$ Problem: Existence of periodicity in the construction of numbers in periodic version of generalized Collatz’ problem and its computational aspects

Abstract

In the paper, another generalization of Collatz's Syracuse problem was discussed. For a given initial integer number, each next integer number is obtained by dividing the previous integer by 2 (T operation), or multiplying it by 3, adding 1 and then dividing by 2 (S operation), or finally, multiplying by 3, adding 2 and then dividing by 2 (V operation), provided that all of these divisions by 2 are possible. The presence of this last operation V makes the problem more general. We ask the following question: How can one find from the given sequence of T, S, V operations whether an initial integer exists which after all these operations, applied in the given order, will end up with the same initial number? We found an algorithm that can quickly find that integer or determine if such an integer doesn't exist. We also discuss relationship of this problem with 3-adic numbers which we use to represent the elements of the periodic sequence.