**ERIC Number:**EJ789617

**Record Type:**Journal

**Publication Date:**2008-Apr

**Pages:**5

**Abstractor:**Author

**Reference Count:**1

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

A Proof of the Collatz Conjecture

Bruckman, Paul S.

International Journal of Mathematical Education in Science and Technology, v39 n3 p403-407 Apr 2008

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.

Descriptors: Equations (Mathematics), Problem Solving, Validity, Mathematical Logic, Mathematical Formulas

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

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**Authoring Institution:**N/A