Cobra Effect · Maths and proof
The margin that was too small
How Andrew Wiles proved Fermat's three hundred year old claim.
7 cards, read aloud in 2:42, with a test and sources.
In the 1630s, Pierre de Fermat wrote a note in the margin of an old book about numbers.
He claimed that no positive whole numbers can make one cube equal to the sum of two cubes, and the same holds for every higher power. He wrote that he had a marvellous proof, which the margin was too small to contain. His son published the note after his death. Fermat’s own copy of the book has not survived.
For more than three centuries, nobody could prove it.
Fermat himself left a method that settles the case of fourth powers. Great mathematicians proved other special cases, but never the whole claim. Most mathematicians now doubt that Fermat really had a correct proof.
At ten years old, Andrew Wiles came across the problem in a library book.
He found it while looking through his local public library. The book told the story of the problem, which looked simple enough for a child to understand. He dreamed that one day he would solve it.
In the 1980s, other mathematicians linked Fermat’s claim to a deep idea about curves.
If a certain conjecture about these curves could be proved, at least for one large family of them, Fermat’s claim would follow. Wiles set to work, largely in secret. He spent about seven years on it.
In June 1993, he announced a proof at the end of a lecture in Cambridge.
It was the last of three lectures at the Isaac Newton Institute. But when the proof was checked in detail, a gap appeared. Wiles struggled for about a year to fix it, joined by his former student Richard Taylor.
In September 1994, he saw how to close the gap.
He later described it as an incredible revelation, and said he stared in disbelief for twenty minutes. The completed proof was published in 1995, in two papers, one of them written with Taylor. In 2016, Wiles received the Abel Prize for the proof.
So when a problem has defeated everyone for centuries, ask what new tools could reach it.
Fermat’s claim was simple to state, but its proof needed twentieth century mathematics. Wiles combined those ideas with years of persistence, and the willingness to repair his own mistake. Some doors open only when the right key has been invented.
Sources
- The Proof, NOVA, PBS. Andrew Wiles tells the story in his own words.
- Fermat’s last theorem, MacTutor History of Mathematics. The history of the problem from Fermat to Wiles.
- The Abel Prize 2016, The Abel Prize. The prize awarded to Wiles for the proof.
Nearby ideas
- The primes never run out. How a proof over two thousand years old shows the primes never run out.
- The limits of proof. How Kurt Godel found limits that no system of proof can escape.
- The long journey of zero. How zero grew from a gap in a number into a number in its own right.
- Seven bridges and a walk that could not be done. How Euler proved a walk impossible without trying a single route.
- Some infinities are bigger than others. How Georg Cantor showed that some infinities are larger than others.