March 31st, 2017, 12:03 AM  #21  
Senior Member Joined: Mar 2017 From: . Posts: 266 Thanks: 4 Math Focus: Number theory  Quote:
Proof to Collatz conjecture. Consider any positive integer n from which the sequence is formed. n has a probability of 0.5 of either being odd or even. If even, we divide it by two. If odd, we multiply it by three, add one and then divide the result by two since the resulting number must be even. This is same as multiplying n by 1.5 and adding 0.5. The resulting integer, say m, will hence either be n/2 or 1.5n+0.5 and also has a probability of 0.5 of being either even or odd. The 0.5 that is added has considerable effect on outcome only if n is 1 (which can explain the repeating cycle when the sequence reaches 1). In this case we will neglect it. Since n has equal chances of being odd or even, it therefore has equal chances of being divided by 2 or being multiplied by 1.5, and so is m and the rest of the outcomes. The factor by which the outcomes are divided by is greater than the factor they are multiplied with and hence, the sequence will converge.  
March 31st, 2017, 09:01 AM  #22 
Senior Member Joined: Aug 2012 Posts: 1,659 Thanks: 427 
I'm stealing this.

March 31st, 2017, 09:34 AM  #23 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,082 Thanks: 2360 Math Focus: Mainly analysis and algebra  The probability that die roll results in an odd number is 0.5, and the probability that that it results in an even number is 0.5. Each of the even numbers 2, 4 and 6 is greater than one of the odd numbers 1, 3 or 5. Your logic claims that if we add odd numbers and subtract evens from a running total we are guaranteed to reach negative numbers at some time. But there are infinite sequences of rolls that do not give us negative numbers (e.g. any sequence containing only odd numbers). Therefore, the probabilistic argument fails. It is thus not robust enough to prove Collatz either. 
April 1st, 2017, 12:11 AM  #24 
Senior Member Joined: Mar 2017 From: . Posts: 266 Thanks: 4 Math Focus: Number theory 
if you choose any integer randomly, it has an equal chance of either being even or odd

April 1st, 2017, 07:38 AM  #25 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,082 Thanks: 2360 Math Focus: Mainly analysis and algebra 
Yes. But there are still infinite sequences of integers that are all odd, that could be chosen randomly.

April 2nd, 2017, 11:38 AM  #26 
Senior Member Joined: Mar 2017 From: . Posts: 266 Thanks: 4 Math Focus: Number theory 
say we randomly pick 1000 integers. there would be around 500 even and 500 odd ones. same as if we flip a coin. In fact the larger the sample space, the more accurate it is. I this case, our sample is infinite coz it consists of all positive integers. It is not possible to have only odd numbers recurring. It is same as saying flipping a coin let's say only a million times and having a million tails. That's impossible.

April 2nd, 2017, 12:11 PM  #27  
Senior Member Joined: Sep 2015 From: USA Posts: 1,649 Thanks: 837  Quote:
it simply has probability $\left(\dfrac 1 2\right)^{1000000}$ quite unlikely yes but it is not 0 probability  
April 2nd, 2017, 12:33 PM  #28  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,082 Thanks: 2360 Math Focus: Mainly analysis and algebra  Quote:
I'm sure, with a little thought you can come up with more situations that need addressing. Perhaps, once you've claimed the prize for your Collatz proof, you can solve these related problems. Last edited by skipjack; April 2nd, 2017 at 12:35 PM.  
April 3rd, 2017, 10:33 AM  #29 
Senior Member Joined: Mar 2017 From: . Posts: 266 Thanks: 4 Math Focus: Number theory  Romsek that's incorrect. It is not under binomial. Take a coin and flip it 50 times. count the number of heads and tails. You will notice it is around 25 give or take a few heads and 25 give or take a few tails. eg maybe 22 heads and 28 tails. it even gets better for 100 flips.. way better for 1000 flips and so forth.

April 3rd, 2017, 06:48 PM  #30 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,082 Thanks: 2360 Math Focus: Mainly analysis and algebra 
What's even better? I think you are wrong anyway. But that needn't worry you as long as you understand why you are so likely to get something close to a 50/50 split. Hint: it's to do with the "probability space". The important thing about the probability space is that is includes plenty of sequences where only 10% or fewer of the flips come up heads.


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