The 3x+1 Problem For Rational Numbers : Invariance of Periodic Sequences in 3x+1 Problem

Abstract

In the paper, we discuss a generalization of 3x+1 Problem sometimes called Collatz’ (Syracuse) Conjecture. For a given initial rational number, each next 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 multiplying by 3, adding 2 and then dividing by 2 (V operation), or finally, just multiplying by 3, and then dividing by 2 (W operation). The presence of the last operation W makes the problem different from what was studied in the previous papers of the author. If a sequence consisted of T, S, V and W operations is given then one can ask whether there is an initial rational number xo for which the application of these operations in the given order will return at the end of this procedure the starting number X0. We will also study how this rational number changes when the order of operations changes. We proved that there is an invariant which is not dependent on the order of operations. In contrast to previous papers, we developed some terminology and notations, and now the results are stated and proved in a reader friendly way. The proofs are also considerably simplified.