Fermat's Last Theorem
Fermat's Last Theorem
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Fermat's Last Theorem, a famous problem in the field of number theory, was conjectured by Pierre de Fermat in the 17th century and remained unproven for over 300 years until it was finally proved by Andrew Wiles in 1994. The theorem states that:
There are no three positive integers a, b, and c , that can satisfy the equation
for any integer value of n greater than 2.
Fermat's Last Theorem can be seen as a natural extension of the Pythagorean theorem. While the Pythagorean theorem states that a2 + b2 = c2 has infinitely many integer solutions (known as Pythagorean triples, like 3, 4, 5), Fermat posited that this relationship breaks down when the power exceeds 2.
Fermat famously claimed in the margin of his copy of Diophantus's "Arithmetica" that he had a marvelous proof of this theorem which the margin was too narrow to contain. This statement by Fermat created one of the most famous problems in the history of mathematics, driving mathematicians for centuries in the search for a proof.
The proof by Andrew Wiles, presented in 1994, is highly complex and uses modern mathematical tools that were not available in Fermat's time, including techniques from algebraic geometry and number theory. The most crucial tool used in Wiles' proof is the modularity theorem for semistable elliptic curves, previously known as the Taniyama-Shimura-Weil conjecture. Wiles' proof of Fermat's Last Theorem was a landmark event in the history of mathematics and was celebrated around the world.
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