Explorations of the Collatz Conjecture - Moravian College

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Chapter 1 Introduction The Collatz Conjecture is a well-known problem that has been puzzling mathematicians for years. The conjecture was first proposed in 1937 by Lothar Collatz, a German mathematician. Since the 1930’s it has been studied in a variety of ways by mathematicians all over the world. It has received other names, such as the Hailstone Numbers, the 3x+1 Problem, the Ulam Conjecture and the Syracuse Problem, from the various mathematicians who have studied it. The past 70 years have led to the discovery of many different properties, extensions and representations of the basic idea of the conjecture. For a nice overview of work that has been done on the Collatz Conjecture, see The 3x+1 problem and its generalizations by Jeffrey Lagarias [5]. Lagarias worked with David Applegate [1, 2] to develop their own ideas and theories on the problem. Their work on trees and total stopping time was of special interest to me, as we will see later on in this paper. Another mathematician whose work was especially helpful, was Riho Terras [6]. He developed an algorithm that I used again and again throughout my re- 6
search. Most of my work, however is an extension of work specifically done by Lynn Garner in On heights in the Collatz 3n+1 problem [4]. This work focuses on the heights (total stopping times) of integers. Guo-Gang Gao [3] produced work similar to Garner’s, but worked with a slightly different function than the one that we used. For the purposes of this research, we will use the representation that is typically known as the 3x + 1 function and is defined as follows. For all positive integers x, ⎧ x ⎪⎨ if x is even 2 T(x) = ⎪⎩ 3x+1 if x is odd . 2 The 3x + 1 conjecture states that all positive integers, when repeatedly iterated through this function, will enter the 2-1 cycle. We say that the integers one and two form a cycle because T(1) = 2 and T(2) = 1. 1.1 Definitions In order to further this discussion of the conjecture, we must define a few terms that are commonly used when discussing this problem. The trajectory of an integer is the sequence that the integer follows through the 3x + 1 function. The trajectory of x can be represented as 〈x, T 1 (x), T 2 (x), ...〉. Then, another way to state the conjecture is that all positive integers have trajectories that contain two. Therefore, we usually look at the trajectory entries just up to the first occurrence of two. For example, the trajectory of 11 is 〈11, 17, 26, 13, 20, 10, 5, 8, 4, 2〉 and the trajectory of 27 is 〈27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 7
Page 1 and 2: Explorations of the Collatz Conject
Page 3 and 4: Dedication This project is dedicate
Page 5 and 6: Preface It all began as a research
Page 7: 6 Future Work 45 A Branch Count Pro
Page 11 and 12: strings to be pairs of parity seque
Page 13 and 14: Figure 2.1: 3x + 1 Tree integer. On
Page 15 and 16: Table 2.1: Total Stopping Times σ
Page 17 and 18: as stems. s i = 0 11...1 01 or s
Page 19 and 20: Now, if we look at the case where x
Page 21 and 22: 3.1 Applying Garner’s Stems In th
Page 23 and 24: Then T 0 (T i+1 (m + 1)) = 3T i+1 (
Page 25 and 26: Chapter 4 Blocks and Strings In the
Page 27 and 28: is an integer if and only if P k (y
Page 29 and 30: Definition 4. A string is a pair of
Page 31 and 32: 4.2 Block and String Structure Sinc
Page 33 and 34: Therefore, v and v ′ are block or
Page 35 and 36: Then, v ′ = 〈v 1 , v 2 , ...v a
Page 37 and 38: We can use Lemma 7, Equation 3.1, T
Page 39 and 40: Since, 2 k ∆ = 2 k − 3 n , 3 k
Page 41 and 42: 2 = 2 k − 2 r 1 1 = 2 k−1 − 2
Page 43 and 44: Chapter 5 Long Runs of Consecutive
Page 45 and 46: can be appended to the sequence in
Page 47 and 48: Chapter 6 Future Work Throughout th
Page 49 and 50: Appendix A Branch Count Procedure b
Page 51: Bibliography [1] David Applegate an

Explorations of the Collatz Conjecture - Moravian College

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