Fermat's Last Theorem - Numberphile

Summary

This script discusses Fermat's Last Theorem, introduced by Pierre de Fermat, who claimed to have a proof that no whole number solutions exist for the equation x^n + y^n = z^n when n is greater than 2. Despite centuries of mathematicians trying to rediscover his proof, it remained elusive until Andrew Wiles, inspired as a child, tackled it through the Taniyama-Shimura conjecture. After years of secret work, Wiles presented a proof, but it initially had an error. With further effort, he corrected it, confirming Fermat’s theorem and solving one of mathematics' greatest challenges.

Takeaways

• 📜 Fermat's Last Theorem was a challenging mathematical problem proposed by 17th-century French mathematician Pierre de Fermat.
• 🧑‍⚖️ Fermat, a judge by profession, made significant contributions to mathematics as a hobby, and one evening he considered the equation x^n + y^n = z^n with n > 2.
• ❌ Fermat claimed to have a proof that there are no whole number solutions to the equation for any n greater than 2, but wrote that his proof could not fit in the margin of the book he was reading.
• 📚 After Fermat’s death, his son published the notes with his 'marvelous proof' claim, sparking centuries of attempts by mathematicians to rediscover this proof.
• 🧐 Despite many efforts over the centuries, no one could find Fermat's original proof, and it's likely that his proof either had a flaw or didn’t exist.
• 👦 The story inspired a young Andrew Wiles, who, at the age of 10, became obsessed with solving Fermat's Last Theorem.
• 🔎 In the 20th century, a connection between Fermat's Last Theorem and the Taniyama-Shimura conjecture was discovered, and Wiles realized solving one could help solve the other.
• 🤫 Wiles worked on this problem in secret for 7 years before presenting his proof, but a mistake was found in the initial version of his proof.
• 🔧 After a year of effort, Wiles, with the help of Richard Taylor, corrected the mistake, ultimately proving Fermat's Last Theorem.
• 🎉 Wiles’ proof became a major mathematical triumph, confirming that Fermat's equation indeed has no whole number solutions for n > 2.

Q & A

What was Fermat's last theorem?

Fermat's last theorem is the statement that there are no whole number solutions to the equation x^n + y^n = z^n, where n is greater than 2.

What was unique about Fermat's note on his theorem?

Fermat claimed to have a 'truly marvelous proof' of the theorem, but stated that the margin of the book he was writing in was too small to contain the proof. He never wrote it down, and it remained a mystery after his death.

Who was Pierre de Fermat?

Pierre de Fermat was a 17th-century French mathematician who worked as a judge by profession. Mathematics was his hobby, and he is known for his work in number theory, including Fermat's last theorem.

How did Fermat's last theorem differ from the Pythagorean theorem?

The Pythagorean theorem involves the equation x^2 + y^2 = z^2, which has many whole number solutions, like 3^2 + 4^2 = 5^2. Fermat's last theorem generalizes this equation for powers greater than 2 and states that there are no whole number solutions for those cases.

Why was Fermat's last theorem considered so difficult to prove?

Although the problem was simple to state, finding a proof was incredibly complex. Mathematicians worked on it for centuries, trying and failing to provide a valid proof, which made it one of the most desirable problems in mathematics.

How did Fermat's last theorem gain public attention?

After Fermat's death, his son published a version of Diophantus' 'Arithmetica,' which included Fermat's marginal notes. One of these notes mentioned the proof of Fermat's last theorem, sparking widespread interest in the mathematical community.

Who ultimately proved Fermat's last theorem?

Andrew Wiles, a British mathematician, proved Fermat's last theorem in the 1990s. After working in secret for seven years, Wiles presented his proof, but it initially contained an error. After a year of collaboration with Richard Taylor, he corrected the mistake.

What role did the Taniyama-Shimura conjecture play in Wiles' proof?

Wiles discovered that proving most of the Taniyama-Shimura conjecture would indirectly prove Fermat's last theorem. This connection motivated him to focus on the Taniyama-Shimura conjecture, which eventually led to his proof of Fermat's theorem.

Why did Wiles work in secrecy on Fermat's last theorem?

Wiles worked in complete secrecy because the challenge was so ambitious and difficult. He wanted to focus entirely on solving the problem without external distractions or pressure.

What was the reaction when Wiles first announced his proof?

Wiles' proof was celebrated worldwide. It made headlines, and Wiles was hailed as one of the greatest mathematicians of the 20th century. However, when an error was found during the checking process, it took him another year to fix the problem and finally complete the proof.