A proof of the Collatz conjecture

• Autores: Paul S. Bruckman
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 39, Nº. 3, 2008, págs. 403-407
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • An elementary proof by contradiction of the Collatz Conjecture (CC) (also known as the '3X + 1' Conjecture), is presented. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N0 is hypothesized; the existence of N0 implies that N0 must generate an infinite sequence {Nk}, each of whose elements is at least as large as N0. A formula for Nk is derived, in terms of an auxiliary sequence {Ek} and the starting value N0. It is shown that each Ek satisfies k = Ek < 1.585k; this, in turn, leads us to conclude that N0 is unbounded, which is a contradiction of its definition, thereby establishing CC.