Explorations of the Collatz Conjecture - Moravian College

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Lemma 2. If x ≡ 2 mod 3, then for all odd integers p ≥ 0, T −1 1 (2p x) Z + and for all even integers q ≥ 0, T −1 1 (2q x) ∈ Z + . Proof. Let x ≡ 2 mod 3. • For all odd integers p ≥ 0, 2 p x ≡ 1 mod 3. Therefore, T −1 1 (2p x) Z + . • For all even integers q ≥ 0, 2 q x ≡ 2 mod 3. Therefore, T −1 1 (2q x) ∈ Z + . □ The tree in Figure 2.1 shows all integers that have a total stopping time between zero and eight. This tree clearly shows us the trajectory followed by each of the integers contained in it. The tree is also helpful to see which integers are at a specific total stopping time. All integers of the same total stopping time line up in a horizontal row. The above lemmas allow us to write an algorithm to count how many integers are at a given total stopping time. The code for this algorithm is found in Appendix A. Using this algorithm, we found N σ , the number of integers with a given total stopping time as shown in Table 2.1, on page 13. In the next chapter, we will take a closer look at integers with the same total stopping time. Specifically, we will turn our focus toward integers that have the same total stopping time as an adjacent integer. 12
Table 2.1: Total Stopping Times σ ∞ N σ σ ∞ N σ 0 1 11 18 1 1 12 24 2 1 13 31 3 1 14 39 4 2 15 50 5 3 16 68 6 4 17 91 7 5 18 120 8 6 19 159 9 8 20 211 10 12 21 282 13
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 and 12: strings to be pairs of parity seque
Page 13: Figure 2.1: 3x + 1 Tree integer. On
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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