Fermat's Last Theorem

Based on Wikipedia: Fermat's Last Theorem

In 1637, a French lawyer scribbled something in the margin of a book that would torment mathematicians for the next 358 years. Pierre de Fermat claimed he had discovered a "truly marvelous proof" of a simple-sounding statement about numbers—but the margin was too small to contain it.

He died without ever writing that proof down. And for more than three centuries, the greatest mathematical minds in history tried and failed to reconstruct what Fermat supposedly knew.

Then, in 1995, an English mathematician named Andrew Wiles finally cracked it—using mathematical tools so advanced that Fermat couldn't possibly have known them. Which raises an irresistible question: Did Fermat actually have a proof? Or was he fooling himself?

The Deceptively Simple Claim

You probably remember the Pythagorean theorem from school. If you have a right triangle, the square of the longest side equals the sum of the squares of the other two sides. Written as an equation: x² + y² = z².

This equation has infinitely many solutions using whole numbers. The most famous is 3, 4, 5—because 9 plus 16 equals 25. These solutions are called Pythagorean triples, and people have known about them for thousands of years. Ancient builders used the 3-4-5 ratio to create perfect right angles.

Fermat's claim was about what happens when you go beyond squares. What if instead of squaring the numbers, you cube them? Or raise them to the fourth power? Or the fifth? Or any power greater than two?

His assertion: there are no solutions. None at all. You will never find three positive whole numbers where a³ + b³ = c³. Or a⁴ + b⁴ = c⁴. Or a⁵ + b⁵ = c⁵. No matter how high you count, no matter how large the numbers you try, you will never find a single example that works.

This is Fermat's Last Theorem. It's called "last" not because it was his final discovery, but because it was the last of his claims that remained unproven.

Why "Theorem" Is a Generous Term

Here's the thing about Fermat: he had a habit of making claims without providing proofs. He would jot down mathematical assertions in letters and book margins, leaving others to verify them later.

Usually, this worked out fine. Other mathematicians would eventually prove his claims, and history would credit Fermat with the discoveries. His theorem about sums of two squares, for instance, was proven decades after his death and bears his name.

But this particular claim was different. Generation after generation of mathematicians threw themselves at it and failed. After a while, people started calling it "Fermat's conjecture" instead of "Fermat's theorem"—a subtle downgrade in mathematical terminology. A theorem is something proven true. A conjecture is something you suspect is true but can't demonstrate.

The distinction matters. In mathematics, you can't just believe something strongly or test it extensively. You need an airtight logical argument that covers every possible case. And there are infinitely many cases to consider here—every combination of whole numbers and every exponent from 3 to infinity.

The Slow March of Partial Progress

Fermat himself proved one case: when the exponent is 4. He showed that x⁴ + y⁴ = z⁴ has no whole number solutions. This wasn't just a random starting point—it was strategically clever. If you can prove the theorem for 4, you automatically prove it for all multiples of 4: the eighth power, the twelfth power, and so on.

A century later, the legendary Leonhard Euler proved the case for exponent 3. Cubes don't work either.

Then progress slowed to a crawl. The case for exponent 5 took until 1825. Exponent 7 wasn't proven until 1839. Each new exponent required fresh insights and increasingly sophisticated techniques.

In the mid-1800s, a German mathematician named Ernst Kummer made a significant breakthrough. He proved the theorem for a large class of exponents called "regular primes." This was real progress—but there were still infinitely many irregular primes that his method couldn't handle.

By the 20th century, mathematicians with computers had verified the theorem for all exponents up to four million. But this was just testing, not proving. Even checking four million cases doesn't prove anything about case four million and one. The mathematical community was stuck.

Enter the Unexpected Connection

The breakthrough came from an unlikely direction: a completely different area of mathematics that seemed to have nothing to do with Fermat's problem.

In 1955, two Japanese mathematicians named Yutaka Taniyama and Goro Shimura proposed a bold conjecture. They suggested that two seemingly unrelated mathematical objects—elliptic curves and modular forms—were secretly the same thing in disguise.

This requires some unpacking. An elliptic curve is a special type of equation that creates a particular shape when graphed. Despite the name, these curves aren't actually ellipses—the terminology is a historical accident. Elliptic curves appear throughout mathematics and have practical applications in cryptography.

Modular forms are something else entirely. They're functions with very specific symmetry properties, living in a different mathematical universe. They're related to number theory and have deep connections to the integers.

Taniyama and Shimura claimed that every elliptic curve corresponds to a modular form. This was an audacious claim, connecting two mathematical continents that no one thought were related. Most mathematicians found it intriguing but considered it hopelessly difficult to prove.

The Bridge to Fermat

For thirty years, the Taniyama-Shimura conjecture sat there, admired but untouchable. Then in 1984, a German mathematician named Gerhard Frey noticed something remarkable.

Suppose Fermat was wrong. Suppose there actually existed three numbers where, say, a⁵ + b⁵ = c⁵. Frey showed that you could use these hypothetical numbers to construct an elliptic curve with very strange properties—so strange that it couldn't possibly have a corresponding modular form.

In other words: if Fermat's Last Theorem were false, then the Taniyama-Shimura conjecture would also be false.

The contrapositive of this statement is what matters: if Taniyama-Shimura is true, then Fermat's Last Theorem must also be true. Proving one would automatically prove the other.

This connection was made rigorous in 1986 by Ken Ribet, building on work by Jean-Pierre Serre. Suddenly, there was a path forward. Fermat's ancient problem was linked to a modern conjecture that people were actively researching. It was still impossibly hard—but at least mathematicians now knew where to aim.

The reaction in the mathematical community was mixed. John Coates, a prominent number theorist, captured the prevailing skepticism:

I myself was very skeptical that the beautiful link between Fermat's Last Theorem and the Taniyama-Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama-Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn't see it proved in my lifetime.

A Secret Seven-Year Quest

Andrew Wiles was ten years old when he first encountered Fermat's Last Theorem in a library book. The problem captivated him. Here was something stated simply enough for a child to understand, yet unsolved after three hundred years. He dreamed of being the one to crack it.

As Wiles grew up and became a professional mathematician, he set aside his childhood obsession. Fermat's Last Theorem had no known path to solution. Working on it seemed like a dead end, a romantic but impractical pursuit.

Then Ribet proved the connection to Taniyama-Shimura, and everything changed.

Wiles had spent his career studying elliptic curves. He understood the mathematical landscape surrounding the Taniyama-Shimura conjecture better than almost anyone. When he learned that proving it would also prove Fermat, he made a fateful decision.

He would work on it in secret.

For seven years, Wiles told almost no one what he was doing. In the competitive world of academic mathematics, this was unusual. Researchers typically share their progress, collaborate, present at conferences. Wiles did none of this. He worked alone in his attic, telling colleagues he was pursuing other projects.

His secrecy was strategic. He didn't want the pressure of public attention. He didn't want competitors racing him to the finish. And frankly, he wasn't sure he would succeed. Better to work in private and emerge with a complete proof than to announce preliminary results and face years of scrutiny.

The Announcement

In June 1993, Wiles delivered a series of three lectures at a conference in Cambridge. The official topic was elliptic curves and related mathematics. But rumors had been circulating. Something big was coming.

By the third lecture, the room was packed. Mathematicians who rarely attended such talks had shown up. Photographers lurked outside.

Wiles proceeded methodically through his argument. He had proven a key case of the Taniyama-Shimura conjecture—specifically, for "semi-stable" elliptic curves. This was exactly the case that Ribet had shown would imply Fermat's Last Theorem.

When Wiles wrote the conclusion on the blackboard, the room erupted. After 358 years, Fermat's Last Theorem was proved. Or so it seemed.

The Gap

The euphoria didn't last.

When Wiles submitted his proof for formal publication, it underwent intense peer review. Expert mathematicians scrutinized every step. And they found a problem.

There was a gap in the argument. One of Wiles's key steps didn't quite work. The logical chain had a weak link.

This is every mathematician's nightmare. Wiles had announced his proof to the world. The media had celebrated. And now it might all collapse.

For months, Wiles struggled to fix the problem. He tried approach after approach, patching and reworking. Nothing worked. The gap seemed unfixable.

At one point, Wiles was ready to give up and publish what he had as incomplete work. Then, in September 1994, he had a breakthrough. Working with his former student Richard Taylor, Wiles found a way around the obstacle. The fix required a different technique than his original approach, but it worked.

The complete proof was published in 1995. Two papers: a 109-page main proof by Wiles, and a shorter companion paper with Taylor addressing the repaired section. Together, they established that the Taniyama-Shimura conjecture was true for semi-stable elliptic curves. And therefore, Fermat's Last Theorem was true.

What Fermat Couldn't Have Known

Wiles's proof is a marvel of modern mathematics. It draws on techniques developed in the 20th century—Galois representations, modular forms, deformation theory, Selmer groups. The mathematical machinery required to make the argument work simply didn't exist in Fermat's time.

This creates an interesting historical puzzle. When Fermat wrote that the margin was too small for his proof, was he lying? Mistaken? Or did he have something else entirely?

The consensus among mathematicians is that Fermat was probably wrong about having a proof. He may have thought he had one, but it likely contained an error he never caught. This happens to mathematicians all the time—a subtle flaw in reasoning that isn't immediately apparent.

Alternatively, Fermat may have had a valid proof for specific cases but incorrectly believed it generalized. His proof for exponent 4 was clever and correct. Perhaps he thought similar techniques would extend to all exponents, not realizing the deeper difficulties involved.

What we can say with certainty is that no proof using only 17th-century mathematics has ever been found. If such a proof exists, it remains hidden four centuries later.

The Broader Impact

Fermat's Last Theorem matters beyond its specific claim. The centuries of failed attempts pushed mathematics forward in unexpected ways.

Ernst Kummer's work on the problem led to the development of algebraic number theory, now a major branch of mathematics. The concept of "ideals" in ring theory—fundamental to modern algebra—emerged from his investigations.

The connection to elliptic curves opened even more doors. Wiles didn't just prove Fermat's Last Theorem; he proved a substantial portion of the Taniyama-Shimura conjecture. Other mathematicians completed the proof in subsequent years. This result, now called the modularity theorem, has become a cornerstone of number theory.

The techniques Wiles developed—particularly his "modularity lifting" methods—have been applied to solve other problems. What began as a 17th-century curiosity became an engine for 21st-century mathematical research.

Recognition

Andrew Wiles received numerous honors for his achievement. In 2016, he was awarded the Abel Prize, one of the highest honors in mathematics. The citation called his proof "a stunning advance" that opened new approaches to numerous other problems.

There's a certain irony in how long recognition took. The Fields Medal, often considered mathematics' equivalent of the Nobel Prize, is only awarded to mathematicians under 40. Wiles was 41 when his corrected proof was published. He missed the age cutoff by a year—though the committee gave him a special silver plaque in acknowledgment of his work.

Perhaps it's fitting. Fermat's Last Theorem resisted solution for 358 years. It makes sense that even the recognition would require patience.

The Elegant Simplicity of the Statement

What makes Fermat's Last Theorem so captivating isn't the proof—which is incomprehensible to all but specialists—but the statement itself. Anyone who understands multiplication and addition can understand what the theorem claims.

Take any three positive whole numbers. Raise each to the same power greater than two. The two smaller results will never sum to the larger one. Never. Not once. Not with any numbers you could ever try.

This is different from saying something is "very rare" or "extremely unlikely." It's an absolute prohibition. Among the infinite possibilities, exactly zero work. The theorem doesn't just say counterexamples are hard to find; it says they cannot exist.

There's something almost unreasonable about this. The Pythagorean equation—the power-of-two case—has infinitely many solutions. The power-of-three case has zero. What changes at that boundary? Why does the mathematical universe suddenly become so uncooperative?

Fermat's Last Theorem doesn't answer these philosophical questions. It simply establishes, beyond all doubt, that the situation is as strange as it appears.

A Connection to Arcadia

Tom Stoppard's play Arcadia features a character obsessed with Fermat's Last Theorem. The play premiered in 1993—the same year Wiles announced his proof. Stoppard had written about an unsolved problem that was solved as his play opened.

In the play, the theorem represents a certain kind of romantic mathematical quest. The search for elegant truth, the willingness to devote years to a single problem, the hope that beauty and logic will ultimately align.

Wiles embodied this romance. A childhood fascination sustained over decades. Years of secret work in an attic. The agony of a flawed proof and the triumph of its repair. His story reads like fiction—but mathematics occasionally produces stories more dramatic than any playwright could invent.

Fermat started it with a marginal note. Wiles finished it with a 109-page proof. Between them stretches 358 years of human effort: wrong turns and partial results, inspired connections and grinding verification, the slow accumulation of mathematical knowledge that eventually made the impossible possible.

The margin, it turns out, really was too small. But not in the way Fermat meant.