Fermat's Last Theorem states that there are no three positive integers $a$, $b$, and $c$ that satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than 2. This was famously conjectured by Pierre de Fermat in 1637 in the margin of his copy of an ancient Greek text, where he claimed he had a proof that was too large to fit in the margin.

The theorem can be formally stated as:

$\text{For } n > 2, \, a^n + b^n \neq c^n \, \text{ for any integers } a, b, \text{ and } c > 0.$

The theorem remained unproven for 358 years and was a major driving force in the development of number theory. It was finally proven by the British mathematician Andrew Wiles, with the help of his former student Richard Taylor, in 1994. Wiles' proof, which involves sophisticated concepts from algebraic geometry and modular forms, was published in 1995.

Here’s a brief overview of some of the key ideas involved in Wiles' proof:

1. Elliptic Curves: These are cubic equations in two variables that have important applications in number theory and cryptography.
2. Modular Forms: These are complex functions that are invariant under certain transformations and are used in number theory.
3. Taniyama-Shimura-Weil Conjecture (Modularity Theorem): This conjecture proposed a deep connection between elliptic curves and modular forms. Wiles' work showed that proving this connection for a special class of elliptic curves would imply Fermat's Last Theorem.

The breakthrough came when Wiles proved that certain elliptic curves could be linked to modular forms, thus confirming a special case of the Taniyama-Shimura-Weil Conjecture and consequently proving Fermat’s Last Theorem.

Would you like more details on any part of the proof, the history of the theorem, or its implications?

Here are some related questions that might interest you:

1. What are elliptic curves and how are they used in number theory?
2. What are modular forms and why are they important in mathematics?
3. What was the Taniyama-Shimura-Weil Conjecture and how does it relate to Fermat’s Last Theorem?
4. How did Andrew Wiles approach the proof of Fermat’s Last Theorem?
5. What challenges did Wiles face during his proof and how did he overcome them?
6. Can you explain the concept of a "proof" in mathematics and why it is important?
7. What other famous theorems in mathematics took a long time to prove?
8. How has the proof of Fermat’s Last Theorem influenced modern mathematics?

Tip: Understanding the proof of a major theorem often requires a deep dive into several complex areas of mathematics. It's helpful to start with the basics and gradually build up your knowledge.