The Bridges to Fermat's Last Theorem - Numberphile

Numberphile・2 minutes read

Pythagorean Theorem and Fermat's Last Theorem are key mathematical concepts, with Fermat's challenge remaining unsolved until Andrew Wiles successfully proved it in 1995 after years of work and collaboration. Wiles' breakthrough changed perspectives on Fermat's Last Theorem's provability and showcased the interconnectedness of mathematical fields.

Insights

  • Fermat's Last Theorem, named after Pierre de Fermat, remained unsolved for centuries due to his claim that no positive integer solutions exist for exponents higher than two, sparking intense mathematical interest and research over the years.
  • Andrew Wiles' groundbreaking proof of Fermat's Last Theorem in 1995, after years of dedication and collaboration, not only resolved a centuries-old mathematical mystery but also highlighted the interconnectedness of different mathematical fields, showcasing the complexity and beauty of the subject.

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Recent questions

  • What is the Pythagorean Theorem?

    The Pythagorean Theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. This fundamental principle has been used for centuries in mathematics and geometry to solve various problems related to right triangles.

  • Who proved Fermat's Last Theorem?

    Andrew Wiles, a mathematician, proved Fermat's Last Theorem in 1995 after years of dedicated work and collaboration with other mathematicians. His proof was a significant breakthrough in the field of mathematics and solved a centuries-old problem.

  • What is the significance of Fermat's Last Theorem?

    Fermat's Last Theorem is a famous mathematical problem that remained unsolved for centuries until Andrew Wiles successfully proved it in 1995. The theorem has had a profound impact on the field of mathematics, leading to the development of new techniques and theories to solve complex equations.

  • How did Andrew Wiles prove Fermat's Last Theorem?

    Andrew Wiles proved Fermat's Last Theorem by working on the modularity theorem, eventually succeeding in 1993. His proof faced challenges initially, but a new insight in 1994 led to a complete proof in 1995, marking a significant achievement in the history of mathematics.

  • What mathematical fields were interconnected in proving Fermat's Last Theorem?

    Andrew Wiles used various techniques, including deformation theory, to prove Fermat's Last Theorem. His work showcased the interconnectedness of different mathematical fields, highlighting the importance of collaboration and innovation in solving complex mathematical problems.

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Summary

00:00

Fermat's Last Theorem: A Mathematical Journey

  • Pythagorean Theorem states that the square of the hypotenuse equals the sum of the squares of the sides of a right triangle, leading to Pythagorean triples.
  • Fermat's Last Theorem involves equations like A cubed plus B cubed equals C cubed and A to the fourth plus B to the fourth equals C to the fourth, with Fermat claiming no positive integer solutions for exponents higher than two.
  • Fermat's marginal notes in Diophantus's text hinted at his inability to prove his assertion for higher exponents, with his son Samuel later publishing these notes.
  • Fermat's Last Theorem was named as such because all of Fermat's previous marginal notes were proven correct, except for the last unexamined assertion.
  • The theorem spurred new mathematical developments over centuries, leading to the creation of various techniques to solve the equation.
  • The modularity conjecture and elliptic curves were linked to Fermat's Last Theorem, with Gerhard Frey proposing a connection between them.
  • Jean-Pierre Serre's letter in 1985 highlighted that proving a small result, epsilon, could imply Fermat's Last Theorem through the modularity conjecture.
  • Andrew Wiles, a graduate student at the time, focused on the epsilon problem, eventually cracking the simplest case of it in May 1986.
  • Wiles' breakthrough led to a collaboration with Barry Mazur, who helped him realize he had already solved the general case of the epsilon problem.
  • Wiles' proof was refined and expanded during a special program at the Mathematical Sciences Research Institute in Berkeley, where he gave lectures and revised his manuscript with input from skeptical colleagues.

13:32

Wiles' Triumph: Proving Fermat's Last Theorem

  • The article by Grothendieck from the 1960s, published in 1990, discusses Fermat's equation and an elliptic curve.
  • The technicalities of the subject are the focus of the article, which received attention in the mathematics community and the press.
  • The Ribet Theorem, named after Ribet, was significant in mathematics and led to a general aversion to naming theorems after oneself.
  • The modularity conjecture, proven by Wiles, implied the truth of Fermat's Last Theorem, changing perspectives on its provability.
  • Wiles, motivated by Fermat's Last Theorem, worked on proving the modularity theorem, eventually succeeding in 1993.
  • Wiles' proof initially faced challenges, but a new insight by Wiles and Taylor in 1994 led to a complete proof in 1995.
  • The proof's completion was a relief, with a gradual understanding of its significance in the mathematics community.
  • Wiles' use of various techniques, including deformation theory, showcased the interconnectedness of mathematical fields.
  • The article on Fermat's Last Theorem, though commonly known by that name, delves into modular forms and presentations, with recognition and pride for the author's role in its discovery.
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