About two countable families in the finite sets of the Collatz Conjecture

Abstract

With t∊ℕ we define the sets Kt and Kt*containing all positive integers that converge to 1 in t iterations in the form of Collatz algorithm. The following are the properties of the { Kt }t∊ℕ and { Kt* }t∊ℕ : countability, empty intersection between the elements of the same family, and - at the end of the work - we conjecture that both of the two families are a partition of . We demonstrate also that each set Kt and Kt* is the union of two sets, a set includes even positive integers, the other, if it’s non-empty, includes odd positive integers different from 1 and we go on proving that the maximum of each set Kt and Kt* is 2t and that Kt ꓵKt*={2t}.

Keywords

• collatz conjecture