September 2nd, 2017, 11:42 PM  #1 
Newbie Joined: Sep 2017 From: United States Posts: 1 Thanks: 0  Collatz Conjecture
I was recently thinking back to the collatz conjecture and decided to just think about it for a while. For those who do not know, the conjecture is basically this: If n is even divide n by 2. If n is odd, multiply n by three then add one. Take that answer and repeat the process such that you have a sequence. For any positive even integer n, does the sequence ever not terminate at 1? My thoughts:
For any positive integer n; If n is even divide n by 2. If n is odd, multiply n by three then add one. Take that answer and repeat the process such that you have a sequence. For any positive integer n, does the sequence ever not contain a value 2^n? 
September 3rd, 2017, 03:25 AM  #2 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26 
That's a lot of words to say something fairly simple. Yes, the 3n+1 step by definition will give you an even number. Yes, when it is 2^a that is the end of the chain as it will go to 1. And yes, you are right, one way to think about attacking the problem is to consider whether all chains starting from b will reach 2^a before they reach b(2^a). But that doesn't actually help very much as this is a horribly difficult thing to prove. Think for instance about the negative loops starting from 5 and 17. Why do those numbers lead to 5( 8 ) and 17(2048 ) rather than to a power of 2? If you could answer that question you might have a start on the problem.

September 4th, 2017, 02:51 AM  #3  
Member Joined: Jul 2014 From: israel Posts: 76 Thanks: 3  Quote:
for any given loop (negative or positive) grater then 3 values the minimum value of that loop will be 12k+7 or 12k+11 12(1)+7=5 12(2)+7=17 you can even go further 96k+7 96k+31 96k+79 96k+91 96k+47 96k+59 96k+71 96k+95 etc...  
September 4th, 2017, 05:37 AM  #4  
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Quote:
What do you mean by minimum value and '3 values' in the sentence above, out of curiosity?  
September 4th, 2017, 10:33 PM  #5  
Member Joined: Jul 2014 From: israel Posts: 76 Thanks: 3  Quote:
4 > 2 > 1 is a cycle with 3 values (2 even and 1 odd) "size" of 3 minimum value is 1 maximum value is 4 What I wrote above is for cycles with "size" of 5 or higher, where the minimum value will be 12k+7 or 12k+11. Last edited by skipjack; September 5th, 2017 at 12:13 AM.  
March 14th, 2019, 05:26 AM  #6 
Newbie Joined: Mar 2019 From: Taiwan Posts: 1 Thanks: 0 
What if instead of an odd number uses prime number then applied 3n+1 to the prime number and uses 2^n+1 you might find the answer


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collatz, conjecture 
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