Explorations of the Collatz Conjecture - Moravian College

More documents

Recommendations

Info

206, 103, 155, 233, 350, 175, 263, 395, 593, 890, 445, 668, 334, 167, 251, 377, 566, 283, 425, 638, 319, 479, 719, 1079, 1619, 2429, 3644, 1822, 911, 1367, 2051, 3077, 1616, 2308, 1154, 577, 866, 433, 650, 325, 488, 244, 122, 61, 92, 46, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2 〉. Using the trajectory, we can obtain the parity vector of an integer. Simply put, the entries of the parity vector are the elements of the trajectory reduced mod 2. We used the notation P n (x) to denote the first n elements of the parity vector of x. We will call this the partial parity vector of length n. For example, P 10 (11) = 〈1, 1, 0, 1, 0, 0, 1, 0, 0, 0〉. The total stopping time σ ∞ (x) of a positive integer x is defined to be the number of iterations needed for that integer to reach one, or ∞ if the trajectory of x does not contain one. In Chapter 2 we investigate the total stopping time through the construction of a graphical representation of the function, which is called a tree. This tree is based upon the inverse map of our function and helps us visualize the function. It also led us to identify several patterns found throughout the integer trajectories. In Chapter 5, we define a run to be a set of consecutive integers that all have the same total stopping time. In order to study these runs we look at some special parity sequences known as blocks, strings and stems. We will define corresponding stems to be a pair of parity sequences of length k such that if two consecutive integers have parity vectors that begin with this pair, then after k iterations, their trajectories merge. See Definition 1 in Chapter 3 for a precise definition. On the other hand, we will define corresponding blocks and corresponding 8
strings to be pairs of parity sequences of length k such that if two consecutive integers have parity vectors that begin with this pair, then after k iterations the next entries in their trajectories are consecutive. See Definitions 2 and 4 in Chapter 4 for further explanation. We use these parity sequences as a tool for investigating the behavior of the 3x + 1 function, including finding long runs of consecutive integers. Finally, in Chapter 6, we discuss what the future might hold for further research on the total stopping times of the Collatz Conjecture. 9
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 and 8: 6 Future Work 45 A Branch Count Pro
Page 9: search. Most of my work, however is
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

Create successful ePaper yourself

Delete template?

Save as template?