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The Collatz Problem

The Collatz Problem. Ellen Dickerson. Introduction to Collatz Conjecture. Named after Luther Collatz Proposed in 1937 Unsolved for over 60 years Also known as a bunch of other things Wondrous numbers 3n+1 conjecture Ulam conjecture And many more. What is the Collatz Conjecture?.

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The Collatz Problem

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  1. The Collatz Problem Ellen Dickerson

  2. Introduction to Collatz Conjecture • Named after Luther Collatz • Proposed in 1937 • Unsolved for over 60 years • Also known as a bunch of other things • Wondrous numbers • 3n+1 conjecture • Ulam conjecture • And many more

  3. What is the Collatz Conjecture? Take any natural number n. If n is even, divide n by 2 to obtain (n/2). If n is odd, multiply it by 3 and add 1 to obtain 3n+1. Repeat the process indefinitely. The Conjecture is that no matter what n you start with, you will always eventually reach 1. ex: try doing this starting with n=3

  4. n=27 • { 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 }

  5. The Collatz conjecture is the process that eventually every number will reach one. If the conjecture is false it can only be because there exists a starting number which gives rise to a sequence which does not contain 1. This sequence might enter a repeating cycles that excludes one, but no such sequence has been found.

  6. Hailstone Pseudocode • function hailstone(n) • while n > 1 • show n • if n is odd then • set n = 3n + 1 • else • set n = n / 2 • endif • endwhile • show n Many numbers have been put into the computer program, and every one has reached the number one.

  7. But what is the Collatz problem? The Collatz conjecture has not been proved nor disproved Every number that has been tried has reached the number one Nobody has been able to prove the Collatz conjecture Although the conjecture has not been proved most mathematicians believe that it is true, because so many numbers have been tried. In fact every number up to 5.764 x 1018

  8. Collatz Conjecture Proofs Indirectly Reverse method: Instead of proving that every natural number leads to one. This method proves that one leads to every natural number ParitySequence: This can be done because when ever n is odd 3n=1 is even.

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