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:

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> Given a positive integer n,

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> If n is even then divide by 2.

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> If n is odd then multiply by 3 and add 1

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> Conjecture: by repeating these operations you will eventually reach 1.

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> Proof:

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> Let n be the smallest positive integer that is a counterexample to the conjecture.

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> If n is even then it can be divided by two to give a smaller number, leading to a contradiction.

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If n is even, then it cannot be the smallest possible

counter-example.

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> Assume n = 4k + 1.

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By definition n (even) = 3k + 1

> Multiply it by 3, add 1, and divide by 2 twice.

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> The result is 3k + 1, a number smaller than n, leading to a contradiction. Therefore n has the form

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> n = 4k - 1.

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> Multiply by 3, add 1, and divide by 2.

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> 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

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> n = 8k - 1

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> Multiply by 3, add 1, and divide by 2.

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> 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.