Event Summary:
Fermat’s Last Theorem claims there are no whole number solutions to the equation xⁿ + yⁿ = zⁿ for any integer n > 2, where x, y, z ≠ 0. Written as a note in the margin of a book in 1637, Pierre de Fermat said he had a “truly marvelous proof,” which was never found. The problem remained unsolved for over 350 years.
Fermat’s Marginal Note (1637):
While reading Arithmetica by Diophantus, Fermat (age 30) scribbled:
“I have discovered a truly marvelous proof… but this margin is too narrow to contain it.”
Equation in Question:
- For n = 2, we have the Pythagorean Theorem:
- x² + y² = z² has infinite integer solutions (e.g., 3² + 4² = 5²).
- For n > 2, Fermat claimed no non-zero integer solutions exist.
Little Proof, Big Challenge:
Fermat never published a complete proof. Mathematicians attempted to prove or disprove the claim for centuries, without success.
Progress Through Special Cases:
- Euler (18th c.): Proved it for n = 3.
- Dirichlet & Legendre: Proved n = 5.
- Lamé and Kummer: Developed algebraic number theory to address higher powers.
The Modern Breakthrough (1994):
Andrew Wiles, a British mathematician, proved the theorem using tools Fermat never could have imagined, including:
- Elliptic curves
- Modular forms
- Ribet’s theorem, which linked the truth of Fermat’s Last Theorem to the Taniyama–Shimura conjecture.
Wiles’ Proof Confirmed:
After a small error was corrected with Richard Taylor’s help, Wiles’ full proof was published in 1995—closing the longest-standing open problem in mathematics.
Why It Matters:
Fermat’s Last Theorem inspired major developments in algebra, number theory, and geometry. Its final proof reshaped modern mathematics.
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