Fermat’s Last Theorem is a statement in number theory that was first proposed by Pierre de Fermat, a French lawyer and mathematician, in 1637. It is one of the most famous theorems in the history of mathematics due to the simplicity of its statement contrasted with the complexity and length of its proof.
Statement of the Theorem
Fermat’s Last Theorem 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.
History
Fermat claimed he had a proof of this theorem but never wrote it down. His note in the margin of his copy of the book “Arithmetica” stated: “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” This claim was made without proof, leading to hundreds of years of attempts by mathematicians to either prove or disprove it.
Proof
The theorem was proved by British mathematician Sir Andrew Wiles in 1994, a full 357 years after it was first proposed. Wiles’ proof was not simple or elementary, but rather involved many complex areas of mathematics, including elliptic curves, modular forms, and Galois representations. His initial submission of the proof in 1993 contained a flaw, but with the help of his former student Richard Taylor, he was able to correct the error and submit the revised proof in 1994.
Impact
Fermat’s Last Theorem has had a profound impact on the field of mathematics. The quest for its proof led to the development of new areas of mathematics and advanced existing ones. Even though the theorem itself doesn’t have many direct applications, the journey to its proof led to advancements that have far-reaching implications in the field.
In summary, Fermat’s Last Theorem is a significant milestone in the history of mathematics. Its proof by Andrew Wiles is considered one of the most notable achievements in the field in the 20th century.