Explorations of the Collatz Conjecture - Moravian College

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Our work with the long runs of consecutive integers in Chapter 5 also has room for research expansion. Do arbitrarily long runs actually exist? Can we make them from any starting parity sequence? Are the majority of the positive integers in a run? Is there any way for us to predict where a run will occur? These ideas, again, also provide areas for exploration. As you can see, there are still so many questions left to answer in regards to this problem. We have only just touched the surface of the work to be done and the results that will come from our attempts to solve this intriguing conjecture. 46
Appendix A Branch Count Procedure branchcount:=proc(b, k, c) # counts the number of integers at height k above an integer b # assuming that b is at height c local count,w,s,v,r; if b=1 or b=2 then # this is taking care of the base case. if k>2 then return(branchcount(4, k, 2)); else return(1); fi; elif b=2ˆc then count:=1; else count:=0; # count the integer 2ˆk # don’t double-count anything else fi; if b mod 3 = 0 then count:=count+0; 47
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 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: Chapter 6 Future Work Throughout th
Page 51: Bibliography [1] David Applegate an

Explorations of the Collatz Conjecture - Moravian College

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