Generalized Collatz Functions: Cycle Lengths and Statistics | Zendy

Hayden R. Messerman | Zendy; Joey LeBeau | Zendy; Dominic Klyve | Zendy

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Generalized Collatz Functions: Cycle Lengths and Statistics

Author(s) -

Hayden R. Messerman,

Joey LeBeau,

Dominic Klyve

Publication year - 2012

Publication title -

international journal of undergraduate research and creative activities

Language(s) - English

Resource type - Journals

ISSN - 2168-0620

DOI - 10.7710/2168-0620.1002

Subject(s) - integer (computer science) , collatz conjecture , mathematics , combinatorics , function (biology) , conjecture , sequence (biology) , number theory , generating function , discrete mathematics , computer science , genetics , evolutionary biology , biology , programming language

Consider the function T (n) defined on the positive integers as follows. If n is even, T (n) = n/2. If n is odd, T (n) = 3n + 1. The Collatz Conjecture states that for any integer n, the sequence n, T (n), T (T (n)), . . . will eventually reach 1. We consider several generalizations of this function, focusing on functions which replace "3n + 1" with "3n + b" for odd b. We show that for all odd b < 400, and all integers n ≤ 106, iterating this function always results in a finite cycle of values. Furthermore, we empirically observe several interesting patterns in the lengths of these cycles for several classes of values of b.

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