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December 30th, 2010, 03:17 AM   #1
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Simple Proof of Fermat's Last Theorem and Beal's Conjecture

Attached is the proof.
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 \'s.doc (30.5 KB, 73 views)

December 30th, 2010, 05:14 AM   #2
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

A few phrases have been added to the new attachment to emphasize that n is an odd integer and A and B do not necessarily need to be integers after the transformation.
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December 30th, 2010, 06:25 AM   #3
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

Quote:
 Originally Posted by MrAwojobi A and B do not necessarily need to be integers after the transformation.
As a result, the transformation fails: you have not shown that FLT implies Tijdeman-Zagier ("Beal"), because FLT relies crucially on the assumption that the numbers are integers. Otherwise there are uncountably many (actually, $\aleph_0\cdot2^{\aleph_0}\cdot2^{\aleph_0}=2^{\ale ph_0}$) counterexamples. For example, $2^3+3^3=(35^{1/3})^3.$

Your FLT proof in the case that n is even is incorrect. In fact, there really isn't a proof at all: you just say that it "collapses to the Pythagorean equation" which, needless to say, does not prove its nonexistence.

Your FLT proof in the case that n is odd comes down to the statement that it "should be clear" that the terms have a common factor. This is not clear, and certainly cannot be assumed here!

In conclusion: Your proof of Beal's conjecture uses an incorrect reduction. Your proof of FLT, in both cases, is by asserting something to be true that immediately gives you the truth of the theorem desired. Of course this is unacceptable; you must prove that these are true, not just claim that they are.

 December 30th, 2010, 07:31 AM #4 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Simple Proof of Fermat's Last Theorem and Beal's Conject Assume, without loss of generality, that FLT is true. QED
December 30th, 2010, 08:35 AM   #5
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

Quote:
 Originally Posted by The Chaz Assume, without loss of generality, that FLT is true. QED
You could have saved Andrew W a ton of time if you had just shown him this in the 80s.

December 30th, 2010, 10:18 AM   #6
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

I have dispensed with requiring n to be odd and have attached a slightly revamped proof.
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December 30th, 2010, 12:12 PM   #7
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

Quote:
 Originally Posted by MrAwojobi I have dispensed with requiring n to be odd and have attached a slightly revamped proof.
This fails for exactly the reasons the earlier proof does: it merely asserts that there is a common factor without proving that there is a common factor (let alone finding it) and it transforms the TZ conjecture into a non-integer version of Fermat (which is solvable!).

 December 30th, 2010, 12:14 PM #8 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Simple Proof of Fermat's Last Theorem and Beal's Conject Would you look over gdmontgomery's FLT proof? I'd really like to know your opinion of it. http://mymathforum.com/viewtopic.php?f= ... 91&p=70114
 December 30th, 2010, 05:07 PM #9 Senior Member   Joined: Aug 2010 Posts: 158 Thanks: 4 Re: Simple Proof of Fermat's Last Theorem and Beal's Conject Just to give more insight into what I am claiming, ponder this simpler but similar problem. As an example, can (a^3)(b^3) be ever equal to 2(c^3)(d^3), for positive values(and not necessarily integers) of a,b,c and d, if a and b don't share a common factor and c and d don't share a common factor? The simple answer is 'no' for the reasons outlined in the proof i.e the similarity of structure of the expressions. The different coefficients is simply what would never make them the same given the conditions above. If you disagree then give a counter example and then I will put this issue to rest.
December 30th, 2010, 09:32 PM   #10
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Re: Simple Proof of Fermat's Last Theorem and Beal's Conject

Did you check that other proof, or at least look it over? I'd really like to know what you think.

Quote:
 Originally Posted by MrAwojobi As an example, can (a^3)(b^3) be ever equal to 2(c^3)(d^3), for positive values(and not necessarily integers) of a,b,c and d, if a and b don't share a common factor and c and d don't share a common factor?
What does "share a common factor" even mean if they're not integers?

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