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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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raycb@live.com

Posts: 41
Registered: 10/25/08
Where is the flaw in this proof of the Collatz Conjecture?
Posted: Jul 25, 2013 10:06 AM
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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.

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.



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