Introduction to Fermat's Last Theorem
Fermat's Last Theorem is one of the best-known mathematical puzzles in history. Pierre de Fermat postulated in 1637 that the equation $ a^n + b^n = c^n $ has no positive integer solutions for $ n > 2 $. This seemingly simple claim defied proof for over 350 years, until it was finally solved in 1994.
The theorem connects number theory, algebraic geometry, and modular forms. Its solution marked one of the most significant achievements in 20th-century mathematics.
Historical Background
Early Attempts
Mathematicians like Euler, Legendre, and Gauss proved special cases in the 1700s and 1800s. By 1980, the theorem was confirmed for all exponents below 4 million, but a general proof remained elusive.
Modular Forms
In 1986, Gerhard Frey proposed a connection between Fermat's Theorem and the Taniyama-Shimura conjecture. This led to the breakthrough that ultimately solved the problem.
Andrew Wiles' Proof
In 1993, British mathematician Andrew Wiles presented a proof at the Isaac Newton Institute, leveraging centuries of mathematical tools. His work unified modularity theorems and elliptic curves—transforming Fermat's cryptic conjecture into a modern mathematical milestone.
“The proof used techniques that would not have been available to Fermat.” — Andrew Wiles
The proof spanned 109 pages and required advanced theories like elliptic curves, modular forms, and Galois representations. A refinement in 1994 fixed an initial flaw, cementing its place in history.
Modern Relevance
Number Theory
The theorem deepened understanding of Diophantine equations and integer solutions across algebraic domains.
Computational Math
Modern cryptographic systems now employ proof techniques developed during the theorem's solution.
Education
Fermat's Theorem remains a canonical example in advanced mathematical education, showcasing persistence and ingenuity.
