Wednesday, June 4, 2014

Daily 9: Fermat's Last Theorem

I watched the video suggested on the course page for my daily work, which provided an overview of proving Fermat's Last Theorem. It's interesting that it took so long to prove a theorem that seems so simple compared to other theorems we have encountered. Further, Fermat claims to have proven it, which seems incredible given that the 20th century proof was very long and technical, using results that were unavailable to Fermat at the time he made the conjecture.

The book I'm reading for the course, The Mathematical Universe by William Dunham, briefly covered Fermat's Last Theorem in a chapter about the 17th century French mathematician. Dunham notes that many people proved Fermat's conjecture was true for specific cases. For instance, Fermat proved that it holds for n = 4, and Euler proved that it holds for n = 3. Dirichlet and Legendre are credited with an 1825 proof that Fermat's conjecture holds for n = 5, Dirichlet a proof for n = 14, and Lame a proof for n = 7. Another proof by Kummer in 1847 proved the conjecture for a large class of exponents. Clearly, people maintained an interest in Fermat's Last Theorem long after it was first proposed.

In the beginning of the 20th century, an offer was made of 100,000 German marks to anyone who provided a correct proof of the conjecture. Of course, many attempts were submitted that were incorrect, and the award seemed a bit less enticing after World War I, when 100,000 marks were equivalent to less than a penny in US dollars. This didn't deter a mathematician by the name of Gerd Faltings, who proved that if the equation held for powers greater than two, then there were at most finitely many solutions. For his find, Faltings won the Fields mMdal in 1986.

Now, enter the work done by Andrew Wiles, whose proof of Fermat's Last Theorem was finally completed in 1995, nearly 360 years after Fermat first made the proposition. The full story given in the documentary is much more exciting than anything I have the willpower to write here, so you should just watch the video linked above for the full story.

What I enjoyed most about this video (and others like it) is that it gave us a look into the people behind the theorem, and all the work and emotion that went into completing the proof. When we read textbooks about math or science, we don't often think of all the work and energy that was put into the results we casually take for granted and use in our own work. Many people dedicate years of their lives to working out one small part of a problem, in the hopes that it can be synthesized with others' results by yet someone else to finally solve a problem. Finally, I thought that it was interesting that Faltings won the Fields Medal for his "partial" proof, yet it has been nearly a decade since Wiles published his proof and has yet to be granted such a prestigious award.

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