B-Theories

The Mathematical Enigma of Fermat’s Last Theorem

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Sushmita PathakB-Theories,,

France, circa 1637. Pierre de Fermat was sitting with his copy of the Arithmetica. Fermat was a lawyer by profession but he had a rather unusual hobby—mathematics. That evening, Fermat was reading a problem on the Pythagoras equation when suddenly he had a moment of revelation. Fermat chuckled to himself and wrote something in the margin of his book:

I have discovered a truly remarkable proof that this margin is too small to contain.

But that wasn’t unusual. In fact, Fermat’s book was filled with such anecdotes. After Fermat’s death, his son published a copy of the Arithmetica with all of his father’s notes. Fermat never actually wrote the proofs but only claimed that he had found them.

Mathematicians all over the world could hardly resist Fermat’s tantalising claims and took up the challenge to find solutions to each of his theorems – And find, they did.

Except for one—the one that Fermat thought was truly remarkable. The one that would haunt mathematicians for the next 358 years.

So, what is this marvellous theorem?

Remember Pythagoras and his famous equation? Simple, wasn’t it? Fermat thought so too and perhaps that was why he decided to spice things up. Take a look at this:

Pythagoras theorem: x2 + y2 = z2

Fermat modified Pythagoras’s equation and replaced the power 2 with a variable integer ‘n’ and claimed that for any value of ‘n’ greater than 2, the equation simply wouldn’t hold true.

Fermat’s last theorem: xn + yn = zn … for n>2

HAS NO WHOLE NUMBER SOLUTIONS!

For a problem that took three and a half centuries to crack, Fermat’s last theorem looks almost childish—a sixth grader could explain it. This is exactly what drove mathematicians crazy. How could an equation that looked so innocent, so simple, be so frustratingly difficult?

The biggest mathematical conundrum

20th-century mathematicians, with calculators and computers, tried to solve the problem by finding a counterexample—a set of numbers that would satisfy the equation and disprove Fermat.

Different permutations and combinations of numbers up to 4 million were checked but not even one satisfied the equation [and numbers are infinite].

Fermat’s last theorem was now the biggest mathematical conundrum.

 

A childhood dream and a lifelong quest

In the mid-twentieth century, while the world’s greatest minds were collectively giving up on Fermat, a young school boy in England called Andrew Wiles walked into his local library and picked up a book called The Last Problem by E. T. Bell. The book was about Fermat’s last theorem.

Andrew was baffled that the world’s biggest riddle was a simple equation that even he could understand. This cruel mathematical irony fascinated him. That day, Andrew promised himself that he would get to the bottom of this mystery.

Inspired by Fermat’s equation, Andrew grew up to become a mathematician. By then, all work on Fermat’s last theorem had come to an impasse.

Andrew Wiles’s childhood dream now seemed more distant than ever.

The Taniyama-Shimura conjecture

Meanwhile, in a land far East, away from Fermat and his theorem, two mathematicians were musing about an altogether different kind of math.

Yutaka Taniyama and Goro Shimura, a Japanese duo from the University of Tokyo put forth an explosive theory in 1955. This theory, called the Taniyama-Shimura conjecture, established a connection between elliptic curves and modular forms—two areas of mathematics that are as different as chalk and cheese.

While elliptic curves can be thought of as structures similar to a doughnut, modular forms are complex functions with an incredibly high level of symmetry [imagine geometric patterns seen through a kaleidoscope].

Taniyama-Shimura claimed that every elliptic curve was also modular, thus relating two fundamentally distinct concepts, but even Taniyama and Shimura had no idea that they had struck gold.

Until 1986.

An unimaginable link

It wasn’t until 1986 that the world understood what the Taniyama-Shimura conjecture really implied. German mathematician Gerhard Frey pointed out a relation between the Taniyama-Shimura conjecture and Fermat’s equation in such a way that proving one would automatically mean that the other was true [think corollary].

The quest for Fermat’s last theorem was rekindled within the mathematical community, the wheels for which had already been set in motion in Japan years ago. Mathematicians were now starting to build on Frey’s approach.

There was light—albeit a tiny speck—at the end of the tunnel.

A dream just within reach

With a probable solution for Fermat’s theorem in the air, Andrew Wiles, the ten-year-old Fermat enthusiast who was now working at Princeton University, could hardly believe his luck. He had been studying about elliptic curves for years now, never quite imagining that it would lead to Fermat’s Last Theorem one day.

From 1986 to 1993, Wiles devoted seven years to the problem. He put aside every other project and embarked upon an almost clandestine mission to make his childhood dream come true. Finally, in a conference in 1993, Wiles presented his proof in a lecture about elliptic curves and modularity.

The mathematics world let out an audible gasp of awe. Math made it to the front pages and Wiles achieved celebrity status. Perhaps the world had started celebrating a bit too early. Things were just about to take a nasty turn for Andrew Wiles and his proof.

The peer review process that follows every mathematical finding found a significant error in Wiles’s theorem. Even though the error did not mean that Wiles’s work was entirely useless, it was still a huge setback. Heartbroken but determined, Wiles set out to do some damage control.

After one year of painstaking work, Wiles was still lost, and now increasingly skeptical too. But you know what they say, spend enough time with a problem and you will eventually find a solution.

On September 19, 1994, suddenly everything fell into place. For Andrew Wiles, this was the perfect Eureka moment. 

What was Fermat’s remarkable proof?

The roundabout solution [Explained in video above] to Fermat’s last theorem stunned mathematicians all over the world.

New questions were now being raised—how could Fermat, a 17th-century mathematician, have possibly come up with this complex proof? Did Fermat actually have a proof or was he just taking us on a wild-goose chase? Maybe Fermat thought he had a proof. Maybe he made a silly mistake. Or maybe there exists another proof that is so simple and elegant that we’re just not smart enough to see.

This still remains a mystery.

The story of Fermat’s equation is so enchanting that it just can’t be confined to the math world. Over the years, it has become the ultimate metaphor for everything enigmatic.

In January 2016, a Super Mario enthusiast created an incredibly challenging Mario level involving a combination of intricate moves with no pauses in between. He called it Fermat’s Last Theorem.

Read the original New York Times article from 1993 that announced Fermat’s Last Theorem had been solved!

In 2016, Andrew Wiles won the Abel Prize for his work with Fermat’s Last Theorem.

Featured Image: By Philippa Warr, via flickr

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