Fermat's Last Theorem: A story of endless pursuit

"I loved every minute of it, however hard it had been"

I have discovered a truly marvelous proof, which this margin is too narrow to contain.

These words were written by the mathematician, Pierre de Fermat, in the margin of a book he was reading, after coming up with a theorem. He died shortly after, and with him, the proof to his theorem, which would remain unsolved for over 350 years!

Pierre de Fermat actually used to work as a judge in 17th Century France and would study Mathematics as a hobby.

One evening, he looked at the famous Pythagoras theorem — x² + y² = z². There are a lot of whole number solutions to this equation.

But what if we change the equation a little. Instead of the squared, what if we changed the exponentials to 3. For example, x³ + y³ = z³.

He couldn’t find any solutions to this. Not only that, but he also believed he had found a proof, that shows that there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

And he wrote in the margin of a book he was reading - Diophantus's Arithmetica —

Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Translation: I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

And then he drops dead.

After his death, his son discovered this book, which was full of such notes. His son published a new version of this book with all his notes. People began looking for these proofs, and in every case where Fermat said he had a proof - there did exist a proof. Except this one secret proof.

It is called Fermat’s Last Theorem because it is the last one which anybody could prove.

But the story has a happy ending.

It starts with a 10-year-old child, Andrew Wiles, who was reading a book one day. The book was The Last Problem by ET Bell. It was all about Fermat’s Last Theorem.

And little Andrew decides he is going to solve that problem. A bright 10-year-old can in fact understand the problem. It's the solution which takes a PhD.

In his late 30s, he comes across Taniyama-Shimura Conjecture (an idea which is not proven), which was proposed in the 50s. Somebody had proven there is a link between this conjecture and the Last theorem. That is, if you could prove the conjecture, it would automatically prove Fermat’s Last Theorem.

Wiles’ childhood passion is reignited. But since, it was such an absurd challenge, Wiles worked on it in complete secrecy for 7 years!

At the end of it, he did find a proof. He became an overnight star in the mathematics world. But someone found a mistake in his proof. Fermat’s Last Theorem was fighting back.

It was a great embarrassment for Wiles.

He worked for another year, with another mathematician, Richard Taylor. Finally, he managed to fix the proof.

Wiles' proof of Fermat's Last Theorem marked a significant milestone in the history of mathematics. It not only solved a centuries-old problem but also demonstrated the power of endless perseverance and the depth of human intellect in unraveling the mysteries of the universe.

Here’s is Andrew Wiles talking about it —