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Topic: Where is the flaw in this proof of the Collatz Conjecture?
Replies: 8   Last Post: Aug 1, 2013 11:58 PM

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Thomas Nordhaus

Posts: 433
Registered: 12/13/04
Re: Where is the flaw in this proof of the Collatz Conjecture?
Posted: Jul 25, 2013 11:09 AM
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Am 25.07.2013 16:06, schrieb raycb@live.com:
> 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.
>
> 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.


So far OK.

>
> 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


Why is this impossible? Let k=2m+1, m odd. Then n = 8m+3. Furthermore
6k-1 = 12m+5 -> 36m+16 -> 9m+4 (odd) > 8m+3 = n. So this is not a
contradiction to n being the smallest counter-example. So m has to be
even. But why should it?

--
Thomas Nordhaus



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