Fermat's Last Theorem and Andrew Wiles

Introduction

• Fermat's Last Theorem: No three positive integers x, y, and z satisfy the equation (x^n + y^n = z^n) for (n > 2).
• Pierre de Fermat: Claimed to have a proof, but it was never found.
• Historic Mystery: Stumped mathematicians for over 300 years.

The Early Puzzle

• Failed Attempts: Notably by Euler for (n = 3).
• Paul Wolfskehl: Offered a reward of 100,000 German marks.

Andrew Wiles' Early Fascination

• Discovery: At age 10 in 1963 in a Cambridge library.
• Dedication: Life-long commitment to solving the theorem.

Key Mathematical Progress

• Elliptic Curves and Modular Forms: Taniyama-Shimura Conjecture states that every elliptic curve has a matching modular form.
• Ken Ribet: Linked the Taniyama-Shimura Conjecture to Fermat’s Last Theorem.

Wiles' Journey

• Graduate Studies: Advised by John Coates to study elliptic curves.
• Secrecy: Worked alone in his attic office for seven years.
• Breakthrough: 1991 conference, learned from student Matthias Flach's work.
• Kolyvagin-Flach Method: Proved helpful but still incomplete.

Final Push

• Collaboration: Confided in Nick Katz in 1993, secret meetings disguised as a course.
• Public Announcement: June 1993 in Cambridge.
• Review: Paper submitted to Inventiones Mathematicae, a flaw discovered.

Resolution

• Richard Taylor: Collaborated in 1994, combined Kolyvagin-Flach and Iwasawa theory by September.
• Publication: Final proof published in 1995.

Reflections

• Personal Achievement: Wiles' greatest triumph.
• Legacy: Solved a mystery that lasted over 300 years.
• Remarkable Determination: Spanning over 30 years.

Inspiration and Resources

• Brilliant: Promotes intuitive learning in math and other disciplines.
• Courses: Geometry, Vectors, Python, etc.