What Are Elliptic Curves and Their Connection to Fermat's Last Theorem?

[Char. Limit]

What are they? And what does it mean to say that all elliptic curves are modular?

Trying to understand Fermat's Last Theorem.

 

[icystrike]

What are they? And what does it mean to say that all elliptic curves are modular?

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?

 

[Char. Limit]

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.

 

[TheForumLord]

I think that if you want us to answer your questions, we need to know what is your mathematical education level...you can also try to read meanwhile Wikipedia

 

[Petek]

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.

 

[Char. Limit]

I think that if you want us to answer your questions, we need to know what is your mathematical education level...you can also try to read meanwhile Wikipedia

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

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.

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

 

[Petek]

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

 

[Char. Limit]

But, if it reduces the possible counterexamples to odd prime exponents, it would seem to rule out *even* numbers greater than 4.

 

[Petek]

I thought that you were claiming that the proof of FLT for n = 4 implied that it held for all even exponents. That's not true. For example, let n = 6. The conclusion that x^6 + y^6 = z^6 has no solutions in integers would follow from the result for n = 3 (because a solution for n = 6 would imply a solution for n = 3 -- (x^2)^3 + (y^2)^3 = (z^2)^3). The fact that there's no solution for n = 4 doesn't help in this case. See the Wikipedia article on FLT for more details. Hope this is clear. If not, please post again.

Petek

 

[Char. Limit]

It's clear. However, my original question was never answered: what are elliptic curves, what is modularity, and why are all elliptic curves modular?

 

[Petek]

It's clear. However, my original question was never answered: what are elliptic curves, what is modularity, and why are all elliptic curves modular?

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

 

[Char. Limit]

Thank you.