Explorations of the Collatz Conjecture - Moravian College

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elif b mod 3 = 1 then w:=floor((k-c)/2); # this is how many branchings will occur above b count:=count+w; if w>0 then for s from 1 to w do count:=count+branchcount((b*(2ˆ(2*s))-1)/3,k,c+2*s); od; fi; elif b mod 3 = 2 then v:=ceil((k-c)/2); count:=count+v; # this is how many branchings will occur above b if v>0 then for r from 1 to v do count:=count+branchcount((b*(2ˆ(2*r-1))-1)/3,k,c+2*r-1); od; fi; fi; return(count); end: 48
Bibliography [1] David Applegate and Jeffrey C. Lagarias, The distribution of 3x + 1 trees, Experiment. Math. 4 (1995), no. 3, 193–209. [2] , Lower bounds for the total stopping time of 3x + 1 iterates, Math. Comp. 72 (2003), no. 242, 1035–1049 (electronic). [3] Guo-Gang Gao, On consecutive numbers of the same height in the Collatz problem, Discrete Math. 112 (1993), no. 1-3, 261–267. [4] Lynn E. Garner, On heights in the Collatz 3n + 1 problem, Discrete Math. 55 (1985), no. 1, 57–64. [5] Jeffrey C. Lagarias, The 3x+1 problem and its generalizations, Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 305–334. [6] Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), no. 3, 241–252. 49
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 and 48: Chapter 6 Future Work Throughout th
Page 49: Appendix A Branch Count Procedure b

Explorations of the Collatz Conjecture - Moravian College

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