#### Date Presented

Spring 4-2021

#### Document Type

Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Dr. Danny Cline

#### Second Advisor

Dr. Kevin Peterson

#### Third Advisor

Dr. Jennifer Styrsky

#### Abstract

The Collatz Conjecture is an unproven problem in mathematics which states that when starting at any positive integer the sequence, for any even-valued element of the sequence, n, the next element is n2, and for any odd-valued element of the sequence, m, the next element is 3m+1, will eventually reach 1. Mod n variations of the Collatz conjecture are sequences constructed such that, for each congruence mod n, the sequence has a separate associated function. The original Collatz Conjecture is a mod 2 sequence as it has a function in which the sequence element, n, is congruent to 0 (mod 2) (n is even) and when n is congruent to 1 (mod 2) (n is odd). The goal of the paper is to show the limitations of John Conway’s theorem on unpredictability when applied to the mod variations of the Collatz Conjecture, explain the causes of these limitations, and show some properties of the cycles of the mod variations. This will be done mainly through the use of computer code written in Python. This research is significant because there has been limited progress toward finding a solution to the Collatz Conjecture by inspecting it directly in the more than 80 years since its introduction. An alternative way of advancing the problem to inspect similar problems, such as mod variations, to see if there are parallels between them and the Collatz Conjecture.

#### Recommended Citation

Murphy, Geoffrey, "Unpredictability and Modular Variations of the Collatz Conjecture" (2021). *Undergraduate Theses and Capstone Projects*. 213.

https://digitalshowcase.lynchburg.edu/utcp/213

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