Exploring Elliptic Curves and Their Relationship to Fermat's Last Theorem

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In summary, the conversation discusses Fermat's Last Theorem and its proof for n=4, which implies that for n>2 there are no integer solutions to the equation a^n+b^n=c^n. The conversation also mentions the use of "infinite descent" in Fermat's proof and the theorem's implications for composite numbers and negative numbers. It is also mentioned that the complete solution involves modularity and elliptic curves. The conversation then delves into a discussion about the mathematical education level of the individuals involved and resources for further understanding. The questions of what elliptic curves are, what modularity is, and why all elliptic curves are modular are raised, but it is noted that these questions do not have easy answers and require
  • #1
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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  • #2
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?
 
  • #3
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.
 
  • #4
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
 
  • #5
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.
 
  • #6
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?
 
  • #7
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
 
  • #8
But, if it reduces the possible counterexamples to odd prime exponents, it would seem to rule out *even* numbers greater than 4.
 
  • #9
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
 
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  • #12
Thank you.
 

1. What are elliptic curves?

Elliptic curves are a type of mathematical curve that follow the equation y^2 = x^3 + ax + b, where a and b are constants. They have a characteristic shape resembling an oval or ellipse, hence the name "elliptic".

2. What are the applications of elliptic curves?

Elliptic curves have numerous applications in fields such as cryptography, number theory, and algebraic geometry. They are commonly used in public key encryption algorithms and digital signatures.

3. How are points on an elliptic curve calculated?

Points on an elliptic curve can be calculated using a process called "point addition", which involves drawing a line between two points on the curve and finding the third point where the line intersects the curve. This process can be repeated to find other points.

4. What is the importance of the "order" of an elliptic curve?

The order of an elliptic curve is the number of points that satisfy the curve's equation. It determines the size of the group of points on the curve, which is important in cryptographic applications as it affects the difficulty of solving the discrete logarithm problem.

5. Are all elliptic curves the same?

No, there are many different types of elliptic curves with varying shapes, sizes, and characteristics. Each curve has its own unique properties and is used for different applications.

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