Fermat's Last Theorem is one of the most famous theorems in the history of mathematics. It states that there are no three positive integers a, b, and c that satisfy the equation:
an + bn = cn
for any integer value of n greater than 2. This theorem was first conjectured by the French mathematician Pierre de Fermat in 1637, who famously noted in the margin of his copy of an ancient Greek text that he had discovered a "truly marvelous proof" of this proposition, but that the margin was too small to contain it.
Historical Background
Fermat's Last Theorem went unsolved for over 350 years, becoming one of the most tantalizing problems in mathematics. Despite Fermat's assertion, no proof was found, and the theorem resisted the efforts of many mathematicians over the centuries. Notable attempts to prove the theorem were made by mathematicians such as:
Leonhard Euler in the 18th century, who was able to prove the case for n = 3 and n = 4.
Joseph-Louis Lagrange, who proved the case for n = 5.
Andrew Wiles, who eventually proved the theorem in 1994.
Wiles' Proof
Andrew Wiles, a British mathematician, announced his proof of Fermat's Last Theorem in 1994, after years of work. His proof is highly complex and uses advanced concepts from algebraic geometry and number theory, particularly the theory of elliptic curves and modular forms.
Wiles' proof involves showing a connection between two seemingly unrelated areas of mathematics: the properties of elliptic curves and modular forms. He proved a special case of the Taniyama-Shimura-Weil conjecture, which posits that every elliptic curve is modular. This connection was crucial, as it allowed Wiles to establish the non-existence of solutions to Fermat's equation for n greater than 2.
Examples
To illustrate Fermat's Last Theorem, consider the following cases:
For n = 2, the equation a2 + b2 = c2 has solutions such as 3, 4, and 5, since:
32 + 42 = 52
For n = 3, while there are integers that satisfy the equation, such as 1, 2, and 3, the equation does not have integer solutions for n greater than 2.
Significance
The resolution of Fermat's Last Theorem has profound implications in number theory and has inspired further research in the field. Wiles' work has led to new developments in the study of elliptic curves and has opened up new avenues of inquiry in mathematics.
References
For more information on Fermat's Last Theorem, you may consider the following resources:
Wikipedia: Fermat's Last Theorem
American Mathematical Society: Fermat's Last Theorem
PBS: Fermat's Last Theorem
In conclusion, Fermat's Last Theorem is a remarkable example of the interplay between simple statements and deep mathematical truths, showcasing the beauty and complexity of mathematics.