The modern story which utimately led to a proof of Fermat's Last Theorem
had its origin with a wonderful conjecture made by the Japanese mathematician
Yutaka Taniyama (1927-1958)
which was later refined by his friend
Goro Shimura (b.1930)
and by the Frenchman André Weil (1906-1998).
It was known that some elliptic curves are
associated with a modular form.
The Taniyama–Shimura–Weil conjecture (or modularity conjecture) said that
all of them were. This is now known as the
Modularity Theorem, which can be stated very concisely:
Every elliptic curve is modular.
Gerhard Frey (b.1944)
connected that to FLT by considering an hypothetical counterexample, for a prime exponent p :
a p + b p = c p
He suspected that the following elliptic curve (the Frey curve)
wouldn't be modular. That would make FLT follow from the modularity conjecture.
y2 = x (x - a p ) (x + b p )
In 1985, Jean-Pierre Serre (b.1926)
analyzed the situation and showed that the validity of Frey's reduction of FLT to the modularity conjecture
was only missing very little (he called that epsilon and that's how the puzzle
became known for a while).
In 1986, Ken Ribet (b.1948)
found a restricted proof of that missing bit, which he was soon able to generalize with some help from
Barry Mazur (b.1937).
From then on, a proof of the modularity conjecture would constitute
de facto a proof of FLT...
This made Andrew Wiles (b.1953) embark on a lonely 7-year
journey to prove the modularity theorem, at least for
elliptic curves, which is enough to prove FLT. Surprisingly enough, he was successful.
The full Modularity theorem was subsequently proved,
using Wiles' own techniques, by
Christophe Olivier Breuil
(b.1968, X1989) and three
former doctoral students
of Andrew Wiles, namely:
Richard Taylor (b.1962),
Fred Diamond (b.1964), and
Brian Conrad (b.1970).
The Bridges to Fermat's Last Theorem (27:52)
by Ken Ribet (Numberphile by Brady Haran, 2015-03-11).
Introducing the proof of Fermat's last theorem (16:06)
by Randell Hyman (2016-09-10).