Explorations of the Collatz Conjecture - Moravian College

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Figure 5.1: First in Pair Flow Chart of g + 1 integers. Using this conjecture, we can construct the flow chart found in Figure 5.1 above. The flow chart shows how to search a parity sequence v to determine whether or not it is the first part of a pair of consecutive integers of the same total stopping time. In our search, we first check the initial entry of our given parity sequence, 〈v〉. If it is a one, we begin searching for blocks that fit the parity sequence. If the initial entry is a zero, we begin searching for strings or stems that fit it. After finding one of our defined special sequences (blocks, strings or stems) that fits the start of the sequence, we check the next entry and continue to move along through the flow chart, searching for the designated special sequences. From our definition of stems, we know to stop once a stem is found because we will now have the first part of a pair. If we run out of entries before reaching a stem, entries 42
can be appended to the sequence in order to make the sequence match the desired special sequence. In these cases, make a stem whenever possible. In order to actually make a new pair, we must then form the parity sequence of the next consecutive integer by using the corresponding sequences in the specified order to the sequence we have just searched. We can then use the work of Terras [6] to find actual integer values to which this sequence corresponds. The basic ideas from Conjecture 3 and Figure 5.1 led us to write a computer program that allowed us to find long runs of consecutive integers of the same total stopping time. In this program, we start with an integer and find how many consecutive integers there are in a row that have the same total stopping time. We do this by calculating and comparing the trajectories of each. While doing the comparison, we keep track of how many steps it takes for the trajectories to coincide. Next, because of our algorithm from Terras [6], we know that if we add an integer multiple of 2 raised to the power of how many steps it takes for the trajectory to coincide to our given integer, we will get another run that will be at least as long as the first. Note that this is assuming that every coinciding pair has to contain a stem. In practice, we have always found this to be true. We can take our new integer and calculate how many consecutive integers of the same total stopping time are less than and greater than our integer, to find the length of our run. We believe that we can continue to do these operations indefinitely. Our only real constraint at the moment is the time it takes the computer to run the calculations. So far, the longest run that we have found is a 39,116-tuple where the first integer in that run is 47, 223, 664, 828, 697, 525, 864, 383 + 5 × 2 274 . 43
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: Chapter 5 Long Runs of Consecutive
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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