Book Review: Fermat's Last Theorem

I am fascinated by how numbers can explain the world, and how mathematical reasoning can explain those numbers. I first discovered this passion by reading Fermat’s Last Theorem, by Simon Singh. This book follows French mathematician Pierre de Fermat’s infamous equation which remained unsolved for centuries before finally being dismantled by Sir Andrew Wiles. I remember sitting through Wiles’ hour-long proof, and although the second slide lost me, my eyes remained glued to the screen. How could such a simple equation cause so much discourse in the mathematical community, even hundreds of years later? More than that, I was in awe of discovering how the roots of what I was learning today had been cemented in history as early as 270 CE by Diophantus in Arithmetica.

Fermat's Last Theorem is a narrative that describes the centuries-long quest to solve the aptly named problem. This theorem was first scribbled in one of Pierre de Fermat's notebooks back in the 17th century and claims that there are no three positive integers a, b, and c such that:

a^n + b^n = c^n

While he claims that such proof holds true, Fermat refuses to prove it. Instead, he teased in the margins of his notebook in Latin: "I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain."

Now if a bold mathematical claim was found in one of my notebooks without a proof attached, I sincerely doubt mathematicians would spend centuries attempting to support my opinion. However, Fermat has a much longer resume than I. Some notable works of his were in number theory, where he popularized proof by infinite descent and managed to get more theorems and numbers named after him. In fact, he was the first to evaluate the integral of general power functions, work that Newton built on when developing the fundamental theorem of calculus. I could write an entire blog post dedicated to Fermat's work, but that is not "integral" to this book review. What is important is the fact that few of Fermat's proofs exist today as publication and status meant nothing to him. Therefore, although frustrating, it was no surprise that the community believed his notebook scrawl, and even less of a surprise that this troublesome theorem did indeed prove true.

Simon Singh's book recounts the tumultuous story from Fermat's 1637 claim all the way to English mathematician Sir Andrew Wiles' eventual proof in 1994. Over the years, many brilliant brains have chipped away at this problem. In the mid-19th century, German mathematician Ernst Kummer was able to prove this theorem for all regular prime numbers. Other mathematicians picked up from here, and eventually the theorem was proved for the first four million prime exponents. If any one of you have taken a real analysis class, however, you know that this still doesn't mean that the four million and first exponent doesn't contradict everything. Therefore, this theorem remained unsolved.

Unrelated, and 100 years later in Japan, mathematicians Goro Shimura and Yutaka Taniyama suspected that a link may exist between modular forms and elliptic curves. Coined the Taniyama-Shimura conjecture, it stood independent of Fermat's theorem and was considered widely important in its own right. It too, however, was considered inaccessible to proof.

I'll skip forward through several more valiant attempts to the late-20th century when one more key discovery was made. German mathematician Gerhard Frey noticed an apparent link between the two aforementioned unrelated and unsolved problems. This link was proven by American Ken Ribet, who in theory claims the following: Any solution that could contradict Fermat's last theorem could also contradict the Taniyama Shimura conjecture.

Now we come back to Sir Andrew Wiles, who had been fascinated with Fermat's theorem for the majority of his life. Hearing that Ribet had proven Frey correct, he decided to attempt to prove Fermat's theorem by proving the Taniyama-Shimura conjecture. In 1993, Wiles proved just enough of Taniyama-Shimura to therefore prove the infamous theorem. After some tricky issues with peer review, the proof was finally accepted in 1995. Simon Singh describes this process in detail, breaking down complex mathematical terms into a well-rounded story for a general audience. It's well written, highly engaging, and captures a feeling of achievement and discovery in mathematical history that ignited my passion, and eventual study, of this wonderful subject. I highly recommend this read.

Can we also touch on how the mathematics used in Wiles' proof did not even exist until hundreds of years after Fermat died? Furthermore, Wiles's proof spans chapters while Fermat considered this too trivial to even write down. Stories like this amaze me, and it will be lost in the pages of history as to whether the French mathematician had a solution to this claim. What we do know, however, is that once something is proven it will always remain that way, and therefore Sir Andrew Wiles will always be the man to take on a legend and win.