Lecture Notes on Fermat's Last Theorem

Introduction

• Fermat's Last Theorem: A famous problem in mathematics with a rich history.
  • Proposed by Pierre Fermat, a French mathematician from the 17th century.
  • Fermat claimed to have a proof, but it was not found in the margins where he wrote it.
  • The claim remained unproven for 350 years.

Historical Context

• Fermat's note was analogous to a "17th century Twitter" note.
• Gained credibility due to Fermat's stature in mathematics.
• Andrew Wiles, a Princeton University mathematician, proved the theorem in 1993, although initially finding a gap in the proof.
• Wiles, with Richard Taylor, completed the proof by 1995.

The Theorem and Its Proof

• Fermat's Last Theorem Statement:
  • For n > 2, there are no whole number solutions to the equation x^n + y^n = z^n.
  • Simple in appearance, as it only involves natural numbers and basic arithmetic.

Connection to Other Mathematical Concepts

• Wiles proved the Shimura-Taniyama-Weil Conjecture, which was linked to Fermat's Last Theorem by Ken Ribet.
• This method is common in mathematics: proving equivalence between different statements to solve a problem.

Fermat's Last Theorem in Comparison

• Compared to Pythagorean Theorem:
  • Pythagorean Theorem: Solutions exist for x^2 + y^2 = z^2 with natural numbers (e.g., 3, 4, 5).
  • Fermat's Last Theorem: No such solutions exist for powers greater than 2.

Drama in Mathematics

• The story of Fermat's Last Theorem is dramatic.
• Involves heartbreak, eventual discovery, and a narrative filled with setbacks and triumphs.
• Highlights the human aspect of mathematical discovery.

Personal Connection

• The speaker shares a personal anecdote about working on a mathematical problem.
  • Experience of working in isolation, like Wiles, and the emotional journey of solving a problem.
  • A sense of ownership over newfound knowledge and understanding.
  • Reflection on the internal and external challenges faced while pursuing a mathematical inquiry.

Conclusion

• Fermat's Last Theorem exemplifies a blend of simplicity and complexity in mathematics.
• The process of solving such problems involves both individual insight and collaborative verification.
• Demonstrates the perseverance and passion involved in mathematical research.