Last Theorem in passing. This is probably one of the few mathematical

theorems so famous that nearly every listener to this podcast has

already heard something about it. As Wikipedia states, this theorem

has a "recognizable status in popular culture", probably because it is

one of the few famous mathematical problems whose statement can be

understood by nearly anyone who has had a basic course in 8th-grade

algebra. But there are a few aspects to the story of this theorem

that are not as commonly known, including the fact that in some sense,

the riddle still hasn't been truly solved. So I thought it would be

fun to talk about it in this week's podcast.

To start with, let's review the basic story of the theorem. In

1637, French number theorist Pierre de Fermat wrote a note in the

margin of his copy of a translation from Diophantus. This note stated

that he had a "truly marvelous" proof that if n is an integer

greater than 2, the equation '(a to the n) + (b to the n) = (c to the

n) has no solutions in which a, b, and c are positive integers. Since

Fermat had a bit of a reputation for stating results that are

ultimately true, and leaving the proof to the reader, this note

attracted a lot of interest. In the century after his death, nearly

all Fermat's results were definitively either proven or disproved, but

this theorem eluded everyone's best efforts, and thus became known as

his "last" theorem. There were many attempts to prove it, especially

after a prize of 100,000 marks was offered in Germany in 1908. Some

have said it bears the distinction of being the single problem with

the greatest number of incorrect proofs published in the history of

mathematics. Over 300 years later, in 1994, Andrew Wiles finally

published a correct proof, and the truth of Fermat's last theorem was

established.

One interesting aspect of the story of this proof is the fact

that Wiles first presented a proof of the theorem in 1993. Wiles'

presentation, at the Isaac Newton institute for Mathematical Sciences

at Cambridge University, was widely hailed for breaking new

mathematical ground in many ways. Newspapers all over the world

mentioned in their headlines that Fermat's centuries-old riddle had

been solved. But, in the months after Wiles' initial presentation, it

became apparent that there was a flaw in the proof, a fact which was

much less widely propagated. In most newspapers, it was a

barely-noticeable correction buried somewhere in their back

sections. I remember having lunch in 1994 with a former college

classmate, who at the time was a Ph.D. student in math, and I was

surprised when he told me the theorem hadn't really been proven! But

fortunately, with the assistance of his student Richard Taylor, Wiles

managed to bridge the flaw in his original proof a year later, and the

theorem is now accepted as proven by the academic community.

So, since a valid proof was finally published, why do I say that

Fermat's riddle hasn't truly been solved? Well, the reason is that

while it is a monumental achievement, Wiles' proof can't possibly be

the one Fermat was mentioning in his margin. It's a massive work

which spans a sort of grand tour of 19th- and 20th- century

mathematics, using numerous concepts that I could not possibly explain

in a podcast, as they are way over my head. But if Fermat truly had a

proof in the 1600's, it would have to be using what we now consider

basic techniques of algebra and number theory, which should be

understandable in modern days even by a high-school student or a

moderately intelligent podcaster. Of course, given the number of

attempts to prove it over the centuries, it's very likely that Fermat

was simply wrong, and the proof he had was actually equivalent to one

of the many invalid proofs published over the years. But the

intriguing possiblity remains: did Fermat have an expressway to

proving this theorem that could bypass the convoluted mathematical

road that Wiles took three centuries later?

And this has been your math mutation for today.

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