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A Proof of the Collatz Conjecture

IJMEST Volume 39, Number 3, ISSN 0020-739X


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 N[subscript 0] is hypothesized; the existence of N[subscript 0] implies that N[subscript 0] must generate an infinite sequence {N[subscript k]}, each of whose elements is at least as large as N[subscript 0]. A formula for N[subscript k] is derived, in terms of an auxiliary sequence {E[subscript k]} and the starting value N[subscript 0]. It is shown that each E[subscript k] satisfies k [less than or equal] E[subscript k] less than 1.585k; this, in turn, leads us to conclude that N[subscript 0] is unbounded, which is a contradiction of its definition, thereby establishing CC.


Bruckman, P.S. (2008). A Proof of the Collatz Conjecture. International Journal of Mathematical Education in Science and Technology, 39(3), 403-407. Retrieved November 19, 2019 from .

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