Editor's note: We are pleased to present the seventh installment in our series, The League of Imaginary Cats. Read more about the Series in our accompanying editorial, and use the navigation links at the top right to catch up.

This is my favourite place. The light is good and the draught is pleasant but not too strong. I have, in fact, been here a good while. My ancestors have always occupied this seat too. The benefits of this locale were discovered by my great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-great-grandfather. 136 generations in this very site. Not many families can boast of such a pedigree, or of such longevity.

The life of one like me is good, if a little hedonistic. I can go anywhere, see everything and disclose nothing. I am the secret keeper, the trusted observer, the familiar.

My ancestors have frequently been painted, exhibited now in many prominent galleries. The poet, John Clare, was enraptured by the thought of cohabiting with us. We are symbols of heroic strength to some, and the enemy to others. Indeed, we even feature in Robert Hooke’s Micrographia, one of the most celebrated scientific books of antiquity.

I also have a sweet tooth – thought you’d like to know.

Today has been a day unlike any other. The visitor to my attic room has been behaving in a most peculiar manner. Usually, its call upon me is for a period of hours each day and is a call of a highly stationary nature. I cannot quite predict the length of each of these calls from one day to the next, but I sense there is a pattern to them. My working hypothesis is that these visits conform to the equation:

One day, I’m sure I’ll discover a truly marvellous proof but find my margin too small to contain it.

Anyway, my visitor has been behaving oddly. Today, it has confounded my hypothesis. I have had not one but several visits, each of an alarmingly short duration. Each follows a most curious pattern. Instead of going to the horizontal plane and shrinking in height and remaining stationary for a while, it goes but does not shrink in height. Instead, it stares at some carbon sheets (not very tasty, I can assure you) on the horizontal plane, bares its teeth in their direction, puts them down, and then leaves the room. Each visit lasts only a few minutes instead of several hours. I am all a-fluster. How can my algebra be so awry?

My visitor has returned. It turned around the room then opened a window. I object! That draught is of greater magnitude than I approve of. How dare you try and dislodge the 136th descendant from this seat!

“What’s that fly doing on the wall?” said Professor Andrew Wiles, rolling up his completed proof of Fermat's Last Theorem.

THWAP!

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Fermat’s Last Theorem

Pierre de Fermat (1607–1665) was a French civil servant with a passion for mathematics. He was notoriously secretive and many of the proofs and techniques he invented were not known until after his death, when his son published his papers. Of these papers, the most famous was that Fermat had discovered a “truly marvellous proof” for the statement

no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two

but that the margin of his book was too small to contain it. This became known as Fermat’s last theorem. Since then, mathematicians had been baffled and defeated in their attempts to find a proof until an English mathematician, Andrew Wiles, spent eight years in almost total isolation secretly working out the proof.

The early stages of Wiles’ proof began in the 1960s when two Japanese mathematicians, Yutaka Taniyama and Goro Shimura, noticed that elliptical equations appeared as if they could be mapped to modular forms. The linking of these two parts of hitherto completely separate branches of mathematics was highly exciting, but no-one could prove the link. This idea that every elliptic equation must be modular became known as the Taniyama-Shimura conjecture.

Next, Gerhard Frey noticed that if Fermat’s equation were re-arranged into an elliptic form, the resulting elliptic form was so weird that it did not have a modular form it could map to. The full proof of the weirdness of the Fermat elliptic equation was provided by Ken Ribet. Therefore, if the Taniyama-Shimura conjecture was true, then Frey’s elliptic equation cannot exist, so there can be no solutions to Fermat’s equation and Fermat’s last theorem is true. This gave Wiles a method for proving Fermat’s last theorem since he merely had to prove the Taniyama-Shimura conjecture. However, proving this conjecture required the development of several new techniques and the modification of many others.

Wiles announced his proof in June 1993, but there was a flaw in the logic that was not solved until October 1994. The completed proof was published in 1995 in a special volume of the Annals of Mathematics. Andrew Wiles has received many honours for his proof, including a knighthood in 2000.

In Fermat’s Last Theorem by Simon Singh, Wiles describes mathematics “in terms of a journey through a dark, unexplored mansion: 'One enters the first room of the mansion and it's dark. Completely dark. One stumbles around, bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, whilst sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of, and couldn't exist without, the many months of stumbling around in the dark that precede them.'” Wiles also noted that, on the day he finally completed his proof, “I walked around the department, and I'd keep coming back to my desk to see if it was still there.” It is this description from Wiles about that one day that forms the basis for the above fancy.

Further reading:

Simon Singh, Fermat’s Last Theorem: the story of a riddle that confounded the world's greatest minds for 358 years (Harper Perrennial, 2005)