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

Abstract

With t∊ℕ we define the sets K_t  and K_t*containing all positive integers that converge to 1 in t iterations in the form of Collatz algorithm. The following are the properties of the { K_t }_t∊ℕ  and { K_t* }_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 K_t  and K_t* 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 K_t  and K_t* is 2^t and that K_t ꓵK_t*={2^t}.