sqishy
October 22nd, 2015, 06:10 PM
I'm doing this thread to show a bit of mathematics that is not well known, yet relatively simple. In this context, a conjecture is a conclusion which has been made on a question, but with only incomplete information. A conjecture is not a proof; a proof is a definite answer, while a conjecture is an educated guess in a way. Lothar Collatz is the person who first put this conjecture out, hence it's called the Collatz Conjecture.
So what is the conjecture about? It is based off two very simple rules.
Take an natural number. Call this number X.
If X is even, divide it by 2. [IF X EVEN: X/2]
If, though, X is odd, then multiply it by 3, then add 1. [IF X EVEN: 3X + 1]
Repeat these steps forever.
It might be intuitive to think that these simple rules will give simple results. They don't, however. Though all natural numbers tested out so far have got to a stage where they reach 4,2,1,4,2,1... in a loop, many numbers take a long and windy journey to get to that, and the length of that journey is not easily relatable to the size of the number you start off with.
For example, if you start with the number 19, it goes as follows:
19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The largest value reached was 88, and the sequence is 21 steps long.
The number 27, though, goes like this:
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.
The largest value reached was way up at 9232, and the sequence is 111 steps long.
255 is the next number which has a larger sequence than 27 when it comes to reaching the looping 4-2-1, and there are bigger and bigger sequences with certain bigger numbers.
The the question behind conjecture is this: Does every natural number, when subject to these rules, eventually reach 1, or does there exist a natural number that never reaches 1 and goes to infinity or some other pattern? The Collatz Conjecture says that every number will reach 1 eventually.
As I said at the start, no proof has been found which confirms or denies this conjecture. Every natural number up to 2 to the power of 60 (also written as 2^60, or just itself as 1,152,921,504,606,846,976) has been tested and shown to eventually reach 1, as of May of this year.
Whether the conjecture is proven or disproven, or if a proof can be found for or against it at all, is yet to be seen.
Just wanted to share this for whoever was interested, or bored, or whatever.
>>> https://en.wikipedia.org/wiki/Collatz_conjecture <<<
http://radar.oreilly.com/wp-content/blogs.dir/2/files/2011/10/1011-collatz-graph.png
So what is the conjecture about? It is based off two very simple rules.
Take an natural number. Call this number X.
If X is even, divide it by 2. [IF X EVEN: X/2]
If, though, X is odd, then multiply it by 3, then add 1. [IF X EVEN: 3X + 1]
Repeat these steps forever.
It might be intuitive to think that these simple rules will give simple results. They don't, however. Though all natural numbers tested out so far have got to a stage where they reach 4,2,1,4,2,1... in a loop, many numbers take a long and windy journey to get to that, and the length of that journey is not easily relatable to the size of the number you start off with.
For example, if you start with the number 19, it goes as follows:
19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The largest value reached was 88, and the sequence is 21 steps long.
The number 27, though, goes like this:
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.
The largest value reached was way up at 9232, and the sequence is 111 steps long.
255 is the next number which has a larger sequence than 27 when it comes to reaching the looping 4-2-1, and there are bigger and bigger sequences with certain bigger numbers.
The the question behind conjecture is this: Does every natural number, when subject to these rules, eventually reach 1, or does there exist a natural number that never reaches 1 and goes to infinity or some other pattern? The Collatz Conjecture says that every number will reach 1 eventually.
As I said at the start, no proof has been found which confirms or denies this conjecture. Every natural number up to 2 to the power of 60 (also written as 2^60, or just itself as 1,152,921,504,606,846,976) has been tested and shown to eventually reach 1, as of May of this year.
Whether the conjecture is proven or disproven, or if a proof can be found for or against it at all, is yet to be seen.
Just wanted to share this for whoever was interested, or bored, or whatever.
>>> https://en.wikipedia.org/wiki/Collatz_conjecture <<<
http://radar.oreilly.com/wp-content/blogs.dir/2/files/2011/10/1011-collatz-graph.png