Orbits Theory. A Complete Proof of the Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.