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

Michele Ventrone


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  and { Kt* }t  : 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}.



Collatz Conjecture

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DOI: http://dx.doi.org/10.23755/rm.v40i1.585


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Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.