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Char. Limit

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Trying to understand Fermat's Last Theorem.

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- Thread starter Char. Limit
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- #1

Char. Limit

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Trying to understand Fermat's Last Theorem.

- #2

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Trying to understand Fermat's Last Theorem.

It should be "every rational elliptic curve is a modular form in disguise" by taniyama-shimura conjecture or modularity theorem. What do you know about Fermat's last theorem?

- #3

Char. Limit

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Finally, I know that the complete solution involves something about modularity and elliptic curves... which I think have the equation y^2=x^3 or something like that.

That's about it.

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- #5

Petek

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I know that it proves that for n>2, there is no integer solution to the equation a^n+b^n=c^n. I know that Fermat proved this for n^4 by using "infinite descent".I know that because of this, his theorem is true for all positive composite numbers. Also, I believe that it would be true for all negative numbers as well, that is, the n>2 could be replaced with |n|>2.

Finally, I know that the complete solution involves something about modularity and elliptic curves... which I think have the equation y^2=x^3 or something like that.

That's about it.

The underlined sentence is false. Fermat's proof of the case n = 4 implies that it suffices to consider odd prime exponents, but not what you typed.

- #6

Char. Limit

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Currently taking AP Calculus BC in high school... also, if something is given to me in understandable terms, i can usually understand it. Usually.

So... does the proof at n=4 prove the theorem for all even numbers greater than 4, maybe?The underlined sentence is false. Fermat's proof of the case n = 4 implies that it suffices to consider odd prime exponents, but not what you typed.

- #7

Petek

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Currently taking AP Calculus BC in high school... also, if something is given to me in understandable terms, i can usually understand it. Usually.

So... does the proof at n=4 prove the theorem for all even numbers greater than 4, maybe?

Sorry, that doesn't follow either.

Petek

- #8

Char. Limit

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- #9

Petek

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Petek

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Char. Limit

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- #11

Petek

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These questions don't have easy answers. The mathematics of FLT lie at the graduate level, if not higher. As suggested earlier in the thread, look at the Wikipedia articles on FLT and elliptic curves. The best elementary introduction to elliptic curves probably is https://www.amazon.com/dp/0387978259/?tag=pfamazon01-20 by Diamond and Shurman. This text covers modularity and such, but isn't an easy read.

HTH

Petek

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- #12

Char. Limit

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Thank you.

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