The Epsilon Conjecture In Fermat's Last Theorem

In summary, the conversation discusses the relationship between Fermat's Last Theorem, the Frey Curve, and the Taniyama-Shimura conjecture. It raises the question of whether disproving the Frey Curve's modularity also disproves the Taniyama-Shimura conjecture and Fermat's Last Theorem. The confusion lies in the uncertainty of the Frey Curve's existence, which is dependent on the truth or falsity of Fermat's Last Theorem.
  • #1
Kevin_Axion
913
2
By supposing there is a solution to Fermat's Last Theorem then according to Frye you can create an elliptic curve that isn't modular. Taniyama-Shimura says that all elliptic curves are modular, so in proving that that Frye curve is not modular which was done by Ribet don't you disprove the Taniyama-Shimura conjecture and Fermat's Last Theorem?

I'm really confused.
 
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  • #2
I think the confusion stems from the idea that the Frey Curve must exist since it could be described and studied. However, its existence depends on whether or not Fermat's Last Theorem is actually true or false. If we do not know whether Fermat's Last Theorem is actually true or false then all we can do is assume it is either true or false which means the existence of the Frey Curve is uncertain. If the existence of the Frey Curve is uncertain, does it make sense to conclude with certainty that the Taniyama-Shimura Conjecture is false if all we do is assume Fermat's Last Theorem is false?
 
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  • #3
Thanks for your answer, I found out my problem elsewhere right after I posted this.
 

FAQ: The Epsilon Conjecture In Fermat's Last Theorem

1. What is the Epsilon Conjecture in Fermat's Last Theorem?

The Epsilon Conjecture is a mathematical conjecture related to Fermat's Last Theorem, which states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The Epsilon Conjecture proposes that if the equation is modified to include a small value of epsilon, where epsilon is a positive real number, then there are infinitely many solutions. This conjecture has not been proven or disproven.

2. Who proposed the Epsilon Conjecture?

The Epsilon Conjecture was proposed by mathematician Roger Heath-Brown in 1994. Heath-Brown is a professor at the University of Oxford and is known for his work in number theory and Diophantine equations.

3. How does the Epsilon Conjecture relate to Fermat's Last Theorem?

The Epsilon Conjecture is a variation of Fermat's Last Theorem, which is one of the most famous unsolved problems in mathematics. The Epsilon Conjecture proposes a modification to the equation in Fermat's Last Theorem, but it still remains unsolved and is not considered a proof of the original theorem.

4. What progress has been made towards proving the Epsilon Conjecture?

There has been some progress made towards proving the Epsilon Conjecture, but it remains an open problem in mathematics. Some mathematicians have been able to find solutions for specific values of epsilon, but a general proof has not been found yet.

5. Why is the Epsilon Conjecture significant?

The Epsilon Conjecture is significant because it is related to Fermat's Last Theorem, which has been an unsolved problem for over 300 years. It also offers a potential avenue for finding solutions to the original theorem, and could potentially lead to a better understanding of number theory and Diophantine equations.

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