Explorations of the Collatz Conjecture - Moravian College

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Chapter 2 Total Stopping Time Recall that 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. In order to get a better visual representation of total stopping time, we can construct a tree to represent the function. Since we conjecture that all positive integers have trajectories that contain 1, the integer 1 will be the root of our tree. In order to generate the tree by starting with 1, we must use the inverse map of our function. We define the inverse map as shown below. T −1 (x) = { 2x, } 2x − 1 3 We will call T −1 0 −1 2x−1 (x) = 2x the even piece of our inverse map and T (x) = the odd piece. On the tree, the even inverse is represented by vertical connections between positive integers. In other words, the integer T −1 0 (x) is drawn directly above the positive integer x. We will refer to these vertical connections as trunks of the tree. New branches coming off these trunks are found using the odd piece of the inverse map. A branch only occurs when T −1 1 (x) is equal to a positive 1 3 10
Figure 2.1: 3x + 1 Tree integer. On the tree, T −1 1 (x) is located above and to the right of x and connected to x with a diagonal line. For example, see Figure 2.1, above. Note T −1 0 (8) = 16 and T −1 1 (8) = 5. Applegate and Lagarias [1] made the following observations; every integer that has a branch coming off it that contains positive integers divisible by 3 is congruent to 5 mod 9 and T −1 1 (x) ∈ Z ⇔ x ≡ 2 mod 3. Then, the following two lemmas detail patterns that are related to these findings. Lemma 1. If x ≡ 0 mod 3, then for all m ∈ Z + , T −1 1 (2m x) Z + . Proof. Let x ≡ 0 mod 3. Then for all positive integers m, 2 m x ≡ 0 mod 3. Therefore, T −1 1 (2m x) Z + . □ 11
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 and 10: search. Most of my work, however is
Page 11: strings to be pairs of parity seque
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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