 ## Tuesday, December 27, 2011

### 34: Not Over Till It's Over

In previous podcasts, you've probably heard me mention Fermat's
Last Theorem in passing.  This is probably one of the few mathematical
theorems so famous that nearly every listener to this podcast has
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
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.

• Timeline of Fermat's Last Theorem
• Fermat's Last Theorem on Wikipedia
• Fermat on Wikipedia