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

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Discussion Overview

The discussion revolves around elliptic curves and their connection to Fermat's Last Theorem (FLT). Participants explore the definitions of elliptic curves, the concept of modularity, and the implications of FLT, particularly regarding integer solutions for various exponents.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants inquire about the nature of elliptic curves and the meaning of modularity, particularly in relation to FLT.
  • One participant asserts that "every rational elliptic curve is a modular form in disguise" according to the Taniyama-Shimura conjecture.
  • Another participant summarizes FLT, stating it proves there are no integer solutions for \( n > 2 \) in the equation \( a^n + b^n = c^n \), and mentions Fermat's proof for \( n = 4 \) using "infinite descent."
  • Some participants express uncertainty about the implications of Fermat's proof for even exponents, questioning whether it extends to all even numbers greater than 4.
  • One participant corrects a claim about the implications of Fermat's proof, stating it only suffices to consider odd prime exponents.
  • Another participant clarifies that the proof for \( n = 4 \) does not imply the theorem holds for all even exponents, using \( n = 6 \) as a counterexample.
  • There is a suggestion that the mathematics involved in FLT and elliptic curves is at a graduate level, indicating the complexity of the topic.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Fermat's proof for even exponents, with some asserting it rules out certain cases while others challenge this interpretation. The discussion remains unresolved regarding the precise nature of elliptic curves and modularity.

Contextual Notes

Participants note that the mathematics of FLT and elliptic curves may require advanced understanding, and some suggest consulting external resources for better comprehension.

Who May Find This Useful

This discussion may be useful for individuals interested in advanced mathematical concepts, particularly those studying number theory, elliptic curves, and their applications in proving theorems like Fermat's Last Theorem.

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

Trying to understand Fermat's Last Theorem.
 
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Char. Limit said:
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?
 
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.
 
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
 
Char. Limit said:
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.
 
TheForumLord said:
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.

Petek said:
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?
 
Char. Limit said:
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
 
But, if it reduces the possible counterexamples to odd prime exponents, it would seem to rule out *even* numbers greater than 4.
 
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
 
  • #10
It's clear. However, my original question was never answered: what are elliptic curves, what is modularity, and why are all elliptic curves modular?
 
  • #11
Char. Limit said:
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
 
Last edited by a moderator:
  • #12
Thank you.
 

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