Can you explain Fermat's Last Theorem?
Fermat's Last Theorem is a mathematical theorem that states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. This theorem was first stated by Pierre de Fermat in the early 17th century, and it remained unproven for over three centuries.

Fermat wrote that he had a proof of the theorem, but he did not write it down, as he was running out of space in the margin of the book he was reading. This led to a long and difficult search for a proof of Fermat's Last Theorem, which was finally discovered by British mathematician Andrew Wiles in 1994.

Wiles' proof of Fermat's Last Theorem was highly complex, and it involved the use of advanced mathematical techniques such as elliptic curves and modular forms. The proof was widely regarded as one of the greatest achievements in the history of mathematics, and it earned Wiles numerous awards and honors.

Overall, Fermat's Last Theorem is a fascinating example of the power of mathematical thinking, and it continues to inspire mathematicians and scientists to this day.
Can you tell me, in layman's terms, how the solution to the Taniyama–Shimura–Weil conjecture, also known as the Modularity Theorem, lead to the proof of Fermat's Last Theorem?
Sure! The Modularity Theorem, which was proved by mathematicians Andrew Wiles and Richard Taylor, showed that every elliptic curve over the rational numbers is modular in a certain technical sense. This meant that elliptic curves could be described using modular forms, which are complex functions that can be thought of as symmetrical shapes.

This result was significant because it allowed Wiles to show that if Fermat's Last Theorem were false, then it would imply the existence of a specific type of elliptic curve that is not modular. However, the Modularity Theorem states that all elliptic curves are modular, so if Fermat's Last Theorem were false, it would contradict the Modularity Theorem.

By showing that Fermat's Last Theorem could not be false without violating the Modularity Theorem, Wiles was able to narrow down the possible ways to prove Fermat's Last Theorem to just a few specific approaches. He then spent several years developing a proof based on these approaches, which ultimately led to the proof of Fermat's Last Theorem.

In other words, the Modularity Theorem provided a crucial piece of the puzzle in the proof of Fermat's Last Theorem, by showing that elliptic curves are modular and helping Wiles to eliminate certain approaches to the problem.