Fermat's Last Theorem, originally known as Fermat's Conjecture, is one of the most famous theorems in the history of mathematics. It was first formulated by the French lawyer and mathematician Pierre de Fermat around 1637. Fermat famously wrote in the margin of his copy of Arithmetica by Diophantus that he had discovered a proof for the theorem, but that the margin was too small to contain it.
Statement of the Theorem
The theorem states that:
There are no whole number solutions to the equation \( x^n + y^n = z^n \) for \( n > 2 \).
Historical Context
- 1637: Fermat jotted down the note in the margin of Arithmetica, claiming to have a proof which was too large to fit.
- 1670: After Fermat's death, his son Samuel de Fermat published the notes, which included the statement of the theorem.
- 17th-19th Centuries: Many mathematicians attempted to prove or disprove the theorem. Notable efforts include those by Leonhard Euler, who proved the case for \( n = 3 \), and Sophie Germain, who made significant progress for certain values of \( n \).
- 1908: Paul Wolfskehl bequeathed a significant sum of money to the one who could prove the theorem, which spurred further interest and research.
The Proof
The theorem remained unproven until 1994 when Andrew Wiles, an English mathematician, announced a proof that utilized techniques from modular forms and elliptic curves. Wiles' proof was based on the Taniyama-Shimura Conjecture, now known as the Modularity Theorem.
- 1993: Wiles presented his proof at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England.
- 1994: After a flaw was discovered in the proof, Wiles, with the help of Richard Taylor, revised and completed the proof, which was published in the journal Annals of Mathematics.
Impact
The proof of Fermat's Last Theorem was a monumental achievement in mathematics:
- It brought together different areas of mathematics including number theory, algebraic geometry, and representation theory.
- It inspired a new generation of mathematicians and popularized the field.
- The theorem's proof has implications in various fields of mathematics beyond just the theorem itself.
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