```Date: Aug 1, 2013 11:58 PM
Author: b92057@yahoo.com
Subject: Re: Where is the flaw in this proof of the Collatz Conjecture?

On Thursday, July 25, 2013 7:06:40 AM UTC-7, ra...@live.com wrote:> The conjecture states that:> > > > Given a positive integer n,> > > > If n is even then divide by 2.> > > > If n is odd then multiply by 3 and add 1> > > > Conjecture: by repeating these operations you will eventually reach 1.> > > > > > > > Proof:> > > > > > > > Let n be the smallest positive integer that is a counterexample to the conjecture.> > > > If n is even then it can be divided by two to give a smaller number, leading to a contradiction.> If n is even, then it cannot be the smallest possible counter-example.  > > Assume n = 4k + 1.> By definition n (even) = 3k + 1 > Multiply it by 3, add 1, and divide by 2 twice.> > > > The result is 3k + 1, a number smaller than n, leading to a contradiction. Therefore n has the form> > > > n = 4k - 1.> > > > Multiply by 3, add 1, and divide by 2.> > > > The result is 6k - 1. If k is odd, then 6k - 1 is one more than a multiple of 4, which is impossible, therefore k is even, and n has the form> > > > n = 8k - 1> > > > Multiply by 3, add 1, and divide by 2.> > > > The result is 12k -1, with k necessarily even. In this manner it can be proved that n must have the form 16k - 1, 32k -1, 64k -1, and so on, requiring n to be infinitely large, which is impossible.
```