BORIS BURDA: How to Prove Fermat’s Great Theorem
Pierre de Fermat (1601–1665) was a French jurist and amateur mathematician. He is known for his contributions that led to modern calculus and his research in number theory. His most famous achievement is Fermat’s Last Theorem. Engraving by an unknown artist, circa 1650 / eisatopon.blogspot.com
ATTENTION — QUESTION!
In 1927, the great mathematician Hilbert was scheduled to give a lecture at the Lorentz Institute. Before his departure, he telegraphed the topic of his lecture: «Proof of Fermat’s Great Theorem». However, he ended up delivering a completely different lecture. Why did he do that?
ATTENTION — CORRECT ANSWER!
If the plane had crashed, everyone would have been convinced that he had proved Fermat’s theorem, which would have brought him immense fame. (Although Hilbert already had plenty of fame, he probably wanted a little more.)
ENTERTAINMENT OF A SUCCESSFUL LAWYER
Let me warn you right away: although this story will touch upon a great mathematical problem, there will be very little actual mathematics in it — just a bit more than in an elementary school course. The thing is this problem, due to its remarkable origin story, has also become a popular concept that is familiar even to those who know nothing about mathematics.
The man who introduced this idea was a fascinating example of a truly fortunate dilettante — someone who excelled in his profession but became famous for a hobby that had nothing to do with his career. This happens more often than one might think: the deaf educator Bell invented the telephone, the painter Morse created his famous code, and there are many other such cases…
The son of a wealthy merchant, Pierre Fermat, received an excellent education, mastered several modern and ancient languages, and even wrote poetry in them. However, he chose to become not a philologist but a lawyer. Fermat simply bought his first position as a royal councilor (something that still happens today, though it was perfectly legal back then) and quickly proved himself worthy of it.
He rapidly advanced in his career, soon becoming a member of the Chamber of Edicts and later even securing a prestigious position as a senior parliamentary judge. However, his legal achievements did not outlive his era, and his mathematical contributions have been remembered forever. It is no coincidence that the word dilettante translates as one who delights — clearly, his leisure pursuits brought him great joy.
THE LONER MATHEMATICIAN
Fermat never attended academies or institutes — simply because they did not exist at the time. He never submitted his work to scientific journals — since those, too, were entirely absent. So, how did his work even reach us? This question applies to other mathematicians of his era as well. It turns out we owe our knowledge of his contributions to yet another dilettante.
His name was Marin Mersenne. A Franciscan monk, he was a teacher and a scientist in his own right (Mersenne primes still play a significant role in fields such as cryptography). In addition, every Thursday, he hosted gatherings for Parisian mathematicians and physicists, providing them with a space to discuss ideas — essentially a home-based academy.
Moreover, Mersenne actively corresponded in Latin with most of the renowned scientists of Europe, keeping them informed about each other’s discoveries. In essence, he served as a one-man scientific journal at a time when such publications did not yet exist. Nearly everything we know about Fermat’s work comes from his correspondence with Mersenne — a fellow dilettante.
However, Fermat’s famous theorem came to light only after his sudden death (he passed away at work during a court session). His son, Clément-Samuel, later published all that remained of his father’s writings, including marginal notes in Diophantus’s Arithmetica. Among those notes was what became known as Fermat’s Last Theorem.
Its statement is remarkably simple: «If n is a natural number greater than 2, then the equation aⁿ + bⁿ = cⁿ has no integer solutions for a, b, and c, other than zero». How did Fermat prove this? The only clue left in the margin was his famous remark: «I have discovered a truly marvelous proof of this, which this margin is too narrow to contain» Did he write it down elsewhere? Was his proof even correct? We will never know.
THE SEARCH FOR THE LOST PROOF
Fermat’s peculiar note didn’t raise too many eyebrows — he had made other marginal comments in books, such as «I could prove this, but I need to feed my cat». However, that particular remark didn’t involve such a sensational claim.
Did Fermat ever write down his proof? A hundred years after his death, Euler requested in a letter that Fermat’s house be searched — perhaps the proof was still there? But nothing was found…
Out of desperation, mathematicians began proving the theorem piece by piece for specific values of n. Fermat himself had left a proof for n = 4, Euler managed to prove it for n = 3, Dirichlet and Legendre did so for n = 5, and Lamé for n = 7… But Fermat’s note suggested that he had proved it in its most general form! Could his proof truly never be replicated?
Many claimed to have succeeded, but there was one problem — errors were found in all of their proofs. There were so many hopefuls that renowned mathematicians grew irritated with enthusiasts requesting reviews of their work, which was often riddled with fundamental mistakes. These amateur theorists even earned a special nickname: «Fermatists».
To defend themselves, several scientific institutions — from the French Academy to the journal Kvant — simply announced that they would not review any claimed proofs of Fermat’s theorem. Such submissions were discarded immediately without being read. However, this policy had little effect on the number of enthusiasts.
Some prominent mathematicians, such as Edmund Landau, even had special response forms printed for Fermatists. The text read: «Dear (blank), Thank you for your submitted manuscript claiming proof of Fermat’s Last Theorem. The first error can be found on page (blank), line (blank)». These forms were filled out by his graduate students.
THE STATUS OF AN UNSOLVABLE PROBLEM
The German mathematician Paul Wolfskehl, who abandoned thoughts of suicide over a lost love after becoming engrossed in yet another article on Fermat’s Last Theorem, established a prize of 100,000 gold marks for solving the problem that had saved his life. After the hyperinflation of 1919, the value of the prize diminished significantly, but thanks to successful investments, it was not lost entirely. However, the question remained — who would be worthy of receiving it?
In Donald Knuth’s famous Mathematical Encyclopedia, which ranked problems by difficulty on a scale from 1 to 50, the highest score — 50 points — was awarded to Fermat’s Last Theorem in the hope that someone might be motivated to solve it. Personally, I was quite surprised to see it there — who could possibly tackle it? After all, it was unsolvable! And I was not alone in thinking so…
The famous astronomer Carl Sagan found a creative use for Fermat’s theorem. Since he was frequently contacted by people claiming to have communicated with extraterrestrials, he proposed a simple test: if these aliens were advanced enough to travel through space, they should certainly be able to prove Fermat’s theorem. Sagan advised these claimants to ask the aliens about it. No convincing responses ever reached him — who knows why.
In the 1930s, Gödel’s incompleteness theorem was proven, demonstrating that mathematics must contain statements that can neither be proved nor disproved. Could Fermat’s Last Theorem have been one of those statements? Personally, I thought it might be. That belief helped me come to terms with the fact that no one had been able to solve it for 300 years…
AND THEN, SUDDENLY…
In 1995, I was visiting a friend — a fairly prominent scientist. During our conversation, he casually mentioned that he had recently attended a seminar where they discussed a proof of Fermat’s Last Theorem, supposedly completed by an Englishman. I asked if they had already found the mistake, to which he hesitantly replied that, apparently, there was no mistake… I must admit, at first, I thought it was a joke.
It turned out to be anything but a joke. The Englishman, Andrew Wiles, had indeed proven it, and in 30 years, no one has found an error in his proof. His work spanned 129 pages and relied on branches of mathematics that Fermat could not have possibly known. If Fermat had truly proved it, he must have done so in some entirely different way — or perhaps he only thought he had proof but actually made an error. Now, we’ll never know…
Wiles was showered with honors, ranging from the prestigious to the downright theatrical. The Queen knighted him, and he rightfully received the remaining Wolfskehl Prize (not a small sum — about £30,000), not to mention the highly respected Abel Prize. They even made a musical about him, in which the crucial step of the proof is supposedly suggested by his wife…
Another great theorem was proven, and another great mystery was solved. Even in the new edition of Knuth’s Encyclopedia, the difficulty of the problem was downgraded — proving it now earns only 45 points instead of 50. The practical benefits of this remain unclear to me, and frankly, life has become less exciting — an enthralling mystery has been replaced by 129 pages of incomprehensible formulas… Things were better before!
