trj
Posts:
95
From:
Korea
Registered:
2/23/12


Re: Where is the flaw in this proof of the Collatz Conjecture?
Posted:
Jul 25, 2013 10:20 AM


> 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. Yes.
> > Assume n = 4k + 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.
Agreed.
> 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,
Why?
> 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
How?
> that n must have the > form 16k  1, 32k 1, 64k 1, and so on, requiring n > to be infinitely large, which is impossible.

