In 1995, Andrew Wiles proved Fermat’s Last Theorem, a problem that had stumped mathematicians for over 350 years.
Never heard of Fermat’s Last Theorem? Well, to explain, let’s start with the Pythagorean Theorem. As you may know, the Pythagorean Theorem states that for right angle triangles, a2+b2=c2, where c is the length of the hypotenuse, and a and b are the lengths of the other 2 sides. We know that there are a number of positive whole number solutions to the Pythagorean Theorem: 3, 4, and 5; 6, 8, and 10; etc.
In 1637, Pierre de Fermat, a French lawyer and amateur mathematician, conjectured that there are no positive whole number solutions for an+bn=cn for values of n>2. So, in other words, there are no positive whole number solutions for a3+b3=c3, or a4+b4=c4, etc. He claimed to have a proof, but did not have room in the margin of the book (Arithmetica) where he wrote this conjecture. For over 350 years, mathematicians tried to discover Fermat’s proof but were unsuccessful. Wiles finally proved the theorem, but he used some modern techniques that would have been unknown to Fermat, which makes ones wonder if Fermat had some other simpler proof that has yet to be discovered.
If you are interested in learning more about Fermat’s Last Theorem, there is a great book and a documentary (both by Simon Singh) that you should check out.
The book, Fermat’s Enigma, provides a detailed and interesting account of Andrew Wiles quest to solve Fermat’s Last Theorem, including explanations of the math appropriate for non-mathematicians. Singh gives us the entire story of Fermat’s Last Theorem starting from Fermat himself and covering the mathematicians along the way that attempted to solve the proof or provided important breakthroughs used by Wiles in his successful proof. Simon Singh wrote the book after producing a documentary about Wiles’ journey for the BBC. The documentary, Fermat’s Last Theorem does not provide as much detail, or course, but I love how, through the interviews with Wiles, you really get a sense of how much of himself he pored into this problem. This problem, which he first learned about from a library book at the age of 10, truly was his life’s work. You can hear the emotion in his voice when he describes how important it was to him and how he felt when he finally completed the proof. The documentary was also produced in the U.S. as The Proof by NOVA, but it isn’t available online. You can see the BBC version in full on YouTube (embedded below).