eprintid: 614 rev_number: 10 eprint_status: archive userid: 2 dir: disk0/00/00/06/14 datestamp: 2022-04-13 23:55:13 lastmod: 2023-07-07 23:30:38 status_changed: 2022-04-13 23:55:13 type: article metadata_visibility: show creators_name: Crespo Álvarez, Jorge creators_id: jorge.crespo@uneatlantico.es title: Orbits Theory. A Complete Proof of the Collatz Conjecture ispublished: unpub subjects: uneat_eng divisions: uneatlantico_produccion_cientifica full_text_status: public abstract: 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. date: 2021-12 publication: Cambridge Open Engage id_number: doi:10.33774/coe-2021-9w3gs refereed: FALSE book_title: Orbits Theory. A Complete Proof of the Collatz Conjecture official_url: http://doi.org/10.33774/coe-2021-9w3gs access: open language: en citation: Artículo Materias > Ingeniería Universidad Europea del Atlántico > Investigación > Producción Científica Abierto Inglés 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. metadata Crespo Álvarez, Jorge mail jorge.crespo@uneatlantico.es (2021) Orbits Theory. A Complete Proof of the Collatz Conjecture. Cambridge Open Engage. (Inédito) document_url: http://repositorio.uneatlantico.es/id/eprint/614/1/orbits-theory-a-complete-proof-of-the-collatz-conjecture.pdf