x^n + y^n = z^n, where n represents 3, 4, 5, ...no solution
"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."
When thinking of mathematics, a few words that come to my mind are equations, symbols, and numbers, but after reading this book, I’ve resolved to add drama and suspense to that list. Indeed the world of math is riddled with dramatic mathematicians who chip away at theorems in their attic to keep from prying eyes, who taunt the world with a proof, only to not write it down anywhere and challenge others to find a proof of their own. From sentencing one to die because they can’t refute a mathematical argument by power of logic, to assuming false identity to facilitate the practice of math, this book is a saga of how mathematicians all across the world teamed up, knowingly or unknowingly–some going as far as fighting against the society–to find a proof for Fermat’s theorem/conjecture that was left unsolved for over three-and-a-half centuries.
Who is Fermat and what is his conjecture?
Born in 1601 France, Fermat wasn’t a mathematician by vocation, instead he was a civil servant who devoured books on math during his free time. Despite having no political ambitions, he climbed the ranks within the parliament because people kept dying from a raging plague. This only helped his venture of studying math since he was expected to maintain a certain degree of isolation from the public due to his position in the civil court so as to not be accused of favouritism.
Apart from this need for isolation, there are other factors that indicate toward his single-minded nature in the pursuit of mathematical knowledge. He refused to socialize with other mathematicians of that age simply because that would waste his time and only serve to be a distraction. If he understood and proved a concept within his mind, that was good enough; he couldn’t care less to explain it to others or answer their nit-picky questions. Fame was no motivator, personal satisfaction was. And this nature of his was what left the world of math confounded for over three centuries ever since he professed in the margins of his textbook that a certain equation had no solutions. There was no proof to this claim, though, just an absolute assertion.
If you recall from grade school, the equation below is taught as Pythagorean theorem or right-angle triangle theorem:
When studying this exposition, Fermat made an ingenious observation and claimed below in the margin.
x^n + y^n = z^n has no whole number solutions for n>2.
A statement so simple, seemingly harmless, and there was no proof for centuries. How it plagued the math community and the history behind the proof that was finally penned down by Andrew Wiles in 1993 and the controversies that came after is what’s covered in Fermat’s Enigma.
Nuances behind proving Fermat’s conjecture
If no one was able to find an example to asset that this claim is false, then it must be true. Right? After all, that’s how scientific theorems are treated; time and again, examples show a certain scientific claim to be true despite there being no logical proof in existence. Think theory of evolution by natural selection, big bang theory etc. There are no step by step logic statements to prove these theorems, just scientific observations that serve to be strong examples to these claims. However, this isn’t the case with mathematical theorems.
So why is it important to have proofs to mathematical theorems?
Let’s say someone assumed that a certain statement is true and used it in their proofs to create other theorems and someone else used this new slew of theorems for their work, and it went on like this until there are hundreds of new equations and thousands of pages of groundbreaking work. What if later down the road, it was proved that the original statement that was assumed to be true was in fact false. All the work that came after falls apart, years wasted since it was all based on a false claim. The book goes over this concept eloquently by comparing this methodology to scientific theorems that come with no absolute proofs and the history of how this blind faith led to some major scientific revolutions.
In fact many have tried to solve Fermat’s theorem. If the proof is to be seen as a race, then there are generations of mathematicians whose contributions shortened this race for future generations by contributing newer methods and newer equations one could use in their proof. That isn’t necessarily to say that the race becomes easy by the end. Mind you, there is a good reason why no one has arrived at a proof for over three centuries.
And many have decidely not worked on Fermat’s theorem. Perhaps for the reasons of self-preservation or for the fear of becoming a victim of sunk cost fallacy, some mathematicians, like Gauss remained ambivalent towards Fermat’s conjecture.
But the one who solved it in the end, Andrew Wiles, and the way he did it, full of secrecy and personal motivations, is in my opinion very fitting to how Fermat made his claim. Both the conception of the equation and its proof, despite being over three centuries apart, are shrouded in drama and intrigue. And it is all revealed in the book.
A personal account on solving a difficult puzzle
I certainly don’t claim to understand the plight of these mathematicians, but while reading this book, I tried to think of whether there was such a problem or puzzle that hounded me for a prolonged period of time. And the answer came immediately.
Over a decade ago, back when I was an undergrad, I used to take a subway train as part of my commute. Right before I went down the stairs to enter the subway, there was a rack full of newspapers for anybody’s taking. Each morning I used to grab one to read a few headlines while all seated in the train, but most importantly to tackle that day’s sudoku in the puzzles section. Most often that not, by the time I arrived at my stop, I would’ve solved the puzzle and recycled the paper on my way out of the station. It was only a matter of time, though, for me to encounter a puzzle of higher difficulty. When I did, I recall tearing that page from the paper and tucking it away in my notes.
I’d then go on staring at that puzzle at all times, during lunch while friends chatted away around me, during lectures while professors went on about math and materials; I was a woman possessed. That puzzle needed solving and I must be the one who did it. I even stopped picking up newspapers from the stand until I finished what I had at hand.
Sometime during my attempts, friends whose interest I piqued would join me in solving it but would ultimately leave thinking it couldn’t be solved. This square grid was nothing like puzzles from other days, it was like trying to climb a smooth, vertical wall with barely any hand or footholds. But one friend stuck around with equal interest in the puzzle. Together, we’d spend all our free time staring at this piece of paper. Then, as we filled in the number one by one, slowly this united effort turned into a competition.
Almost a month later, by the time we were close to finishing that puzzle, there was a silent understanding: it wasn’t a collaborated effort anymore, instead it had become a race. A race to see who got the last number in. By then the logic was solved, all that was left was to fill in the blanks. Our eyes darted all over for that missing number in a row or column. I recall putting a pause on writing down these numbers because that took away precious seconds from my search time, so instead I used my fingers as placeholders and tried to memorize what went in the blank. It was a rush, my heart racing those last few moments. And when it was done at last, there was this moment of elation, of having accomplished something even if it was of no consequence in the long term or even if it had nothing to contribute to the world.
I imagine proving Fermat’s conjecture must’ve been something similar but at a significantly larger scale. Mathematicians who shifted their efforts to other things were like my friends who stopped working on the puzzle with me, and the one who stuck around had faith in our combined effort. This book is magical in the sense how it makes this esoteric plight relatable to a non-mathematician.
A brief history that led to solving the theorem
For an example of the kind of history that is covered in the book that led to the ultimate proof of the theorem, here’s a few snippets from, in my opinion, one of the most captivating periods:
In 332 B.C., having conquered Greece, Asia Minor, and Egypt, Alexander the Great decided that he would build a capital that would be the most magnificent in the world. Alexandria was indeed a spectacular metropolis but not immediately a center of learning. It was only when Alexander died and Ptolemy I ascended the throne of Egypt that Alexandria became home to the world’s first-ever university. Mathematicians and other intellectuals flocked to Ptolemy’s city of culture, and although they were certainly drawn by the reputation of the university, the main attraction was the Alexandrian Library.
…
Even tourists to Alexandria could not escape the voracious appetite of the Library. Upon entering the city, their books were confiscated and taken to the scribes. The books were copied so that while the original was donated to the Library, a duplicate could graciously be given to the original owner.
Why is information about this library important to the context of Fermat’s conjecture? Alexandria is where mathematics flourished for as long as the university and the library existed. The mathematical discoveries that took place in these hallowed halls are what led to Fermat’s ingenious assertion. All this is covered in the book, from what happened at the Alexandrian University to the disastrous events that caused its downfall close to a millenia later.
Many such fascinating accounts of history are covered in this book, along with clear and digestible explanations of math and how it connects to the proof written by Andrew Wiles. Even the reveal of the proof is one dramatic escapade and I thoroughly enjoyed each moment of it.