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March 5th, 2018, 06:34 PM   #11
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Admittedly off topic. I have long wondered what attracts people to things like Fermat's Last Theorem.

Part, I suspect, is an unwarranted intuition that things easy to express are easy to prove. Part is a simple desire to solve a puzzle; I have to believe that desire motivated Wiles and the many skilled mathematicians who had sought a proof before him. And for many decades, I am confident that people did not understand the actual complexity of the problem.

Now that a proof has been found and the complexities involved in finding a proof have been delineated, the continuing interest baffles me. "Wiles found a proof and x years later I did too" seems a rather puerile goal. Or am I missing something?
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March 19th, 2018, 06:31 AM   #12
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Math Focus: Number theory, Applied maths
See below for interesting results of further extensive research. Comments welcome.


Although it is known that Fermat’s last theorem is true for all powers higher than two, an attempt to disprove this with odd powers, might be to show that there is at least one set of three integers A>B>C where all of the following statements are true :-
1. An MOD (B +C) =0
2. Bn MOD (A-C) = 0
3. CnMod(A-B) = 0
4. (Bn+Cn) MOD A =0
5. (An-Bn) MOD C= 0
6. (An -Cn) MOD B = 0
7. A, B & C all different
8. A, B & C relatively prime
9. A.B.C MOD 2 =0
10. When 2n+1 is prime then A.B.C MOD (2n +1) = 0 (Sophie Germain, but under Nom de plume until 1807)
11. A,B.C not in arithmetical progression (Bottari (1907), Goldziher (1913), Milhajinee (1952), Rameswar Rao (1969) Alternatively if it can be shown that for all possible sets of A,B & C, at least one of these statements cannot be true, then we would have a new proof of the theorem.
However a comprehensive computer aided examination of all 35,820,200 possible sets of A,B & C from 3,2,1 to 600,699,698 and powers from 3 to 21 have shown that all the above statements are true in the number of triads as follows:-
Power 3 20,522
Power 5 13,659
Powers 7, 11, 17, 19 54,749 (not statement 10)
Power 9 8,245
Power 11 6,892
Power 15 5,207
Power 21 3,702
(The sets of triads where 2p+1 is not prime are identical , and of which, the others subsets.)

For comparison the number or primitive Pythagoras triples up to A,B, C = 593,465,368 is only 95. When all statements other than 1 are true, (statement 11 with A,B,C = 5,4,3)
So how can that be since FLT is true? The answer is that there is one more statement that is true which confirms that FLT is true (Apart from A^n - B^n - C^n >< 0).
I leave it to the reader to discover that other statement. Clue : check ONE more odd prime power.

Last edited by magicterry; March 19th, 2018 at 06:38 AM. Reason: update
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March 19th, 2018, 02:16 PM   #13
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Originally Posted by magicterry View Post
For comparison the number of primitive Pythagoras triples up to A,B, C = 593,465,368 is only 95.
Am I reading this correctly? You are saying that there are only 95 primitive Pythagorean triples with hypotenuse $\displaystyle \leq$ 593,465,368?
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March 19th, 2018, 02:45 PM   #14
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Originally Posted by SDK View Post
in the hopes it penetrates the crazy.
I think the 2016 election demonstrated that "the crazy" is weaker only than adamantium and thus is virtually impenetrable.
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March 19th, 2018, 03:21 PM   #15
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Originally Posted by romsek View Post
I think the 2016 election demonstrated that "the crazy" is weaker only than adamantium and thus is virtually impenetrable.
I had to look that up. Adamantium is "a fictional metal alloy appearing in American comic books published by Marvel Comics. It is best known as the substance bonded to the character Wolverine's skeleton and claws."

Today I learned!

What do you think is crazy? That Trump became president? Or that the only viable options available to the American people were Trump and Hillary? Or that the DNC rigged its own process to nominate the only person in the country who could manage to lose to Trump? I'd go with all of the above.
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