Step 1) If the number is even, cut it in half; if the number is odd, multiply it by 3 and add 1. :). I've just written a simple java program to print out the length of a Collatz sequence, and found something I find remarkable: Consecutive sequences of identical Collatz sequence lengths. The sequence is defined as: start with a number n. The next number in the sequence is n/2 if n is even and 3n + 1 if n is odd. An Automated Approach to the Collatz Conjecture. Lothar Collatz, two years after receiving his doctorate, introduced the idea of a conjecture in 1937. And the conjecture is possible because the mapping does not blow-up for infinity in ever-increasing numbers. The Collatz conjecture simply hypothesizes that no matter what number you start with, youll always end up in the loop. As an example, 9780657631 has 1132 steps, as does 9780657630. Double edit: Here I'll have the updated values. But I've only temporarily time, due to familiar duties @DmitryKamenetsky you're welcome. [14] Hercher extended the method further and proved that there exists no k-cycle with k91. I had to use long instead of int because you reach the 32bit limit pretty quickly. Are computers ready to solve this notoriously unwieldy math problem? As of 2020[update], the conjecture has been checked by computer for all starting values up to 268 2.951020. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go. I do want to know if there exist a longer sequence of consecutive numbers that have the same number of steps, $$\frac{3^i}{2^k}\cdot n_0+(\frac{\delta}{2^k})=1$$, $$\frac{2^{k-1}}{3^i}1$ such that the number of Collatz steps needed for $238!+m$ to reach $1$ differs from that for $238!+1$. The Collatz conjecture is one of the great unsolved mathematical puzzles of our time, and this is a wonderful, dynamic representation of its essential nature. Repeat this process until you reach 1, then stop. I've regularly studied sequences starting with numbers larger than $2^{60}$, sometimes as large as $2^{10000}$. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7107 (for x = 9663). Applying the f function k times to the number n = 2ka + b will give the result 3ca + d, where d is the result of applying the f function k times to b, and c is how many increases were encountered during that sequence. and Applications of Models of Computation: Proceedings of the 4th International Conference By rejecting non-essential cookies, Reddit may still use certain cookies to ensure the proper functionality of our platform. The section As a parity sequence above gives a way to speed up simulation of the sequence. for the mapping. The initial value is arbitrary and named $x_0$. Anything? This is An equivalent form is, for If n is odd, then n = 3*n + 1. The 3n+1 rule is iterated through 36 times, so this graph is incomplete for larger numbers. If the integer is even, then divide it by 2, otherwise, multiply it by 3 and add 1. It was the only paper I found about this particular topic. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Look it up ; it's related to the $3n+1$ conjecture (or the Collatz conjecture), and the name is not irrelevant. These two last expressions are when the left and right portions have completely combined. Privacy Policy. Then in binary, the number n can be written as the concatenation of strings wk wk1 w1 where each wh is a finite and contiguous extract from the representation of 1/3h. There's nothing special about these numbers, as far as I can see. 2 Figure:Taken from [5] Lothar Collatz and Friends. We can trivially prove the Collatz Conjecture for some base cases of 1, 2, 3, and 4. [21] Simons (2005) used Steiner's method to prove that there is no 2-cycle. To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer).
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