There are lots of Pythagorean triples; triples
of whole numbers which satisfy:
x^{2} + y^{2} = z^{2}.
But are there any which satisfy
x^{n} + y^{n} = z^{n},
for integer powers n greater than 2?
The French jurist and mathematician
Pierre de Fermat claimed the answer was "no",
and in 1637 scribbled in the margins of a book
he was reading (by Diophantus) that he had
"a truly marvelous demonstration of this proposition
which the margin is too narrow to contain".
This tantalizing statement (that
there are no such triples) came to be
known as Fermat's Last Theorem even though it
was still only a conjecture, since Fermat never
disclosed his "proof" to anyone.
Many special cases were established, such as for
specific powers, families of powers in special cases.
But the general problem remained unsolved for centuries.
Many of the best minds have sought a proof of this
conjecture without success.
Finally, in the 1993, Andrew Wiles, a mathematician
who had been working on the problem for many years,
discovered a proof that is
based on a connection with
the theory of elliptic curves (more below).
Though a hole in the proof was discovered,
it was patched by Wiles and Richard Taylor in 1994.
At last, Fermat's conjecture had become a "Theorem"!
Presentation Suggestions:
Students often find it amazing that such a
great unsolved problem in mathematics can be so simply
stated.
Often they don't realize that mathematics,
like other disciplines, has unsolved questions that
spur on the development of new ideas.
The Math Behind the Fact:
Pursuit of this problem and related questions has
opened up new fields of number theory and connected
it with other fields, such as the theory
of elliptic curves.
Wiles' based his work on a 1986 result of Ken Ribet
which showed that the
TaniyamaShimura conjecture in arithmetic/algebraic
geometry implies Fermat's Last Theorem. Wiles was
able to prove the TaniyamaShimura conjecture,
which establishes a "dictionary" between
elliptic curves and modular forms,
by converting elliptic curves into
something called Galois representations.
This way of thinking
brought new techniques to bear on a centuriesold
problem.
By the way...
given the depth of techniques that were eventually
necessary to push the proof through, it is widely
believed that Fermat was mistaken in thinking he had
a proof.
How to Cite this Page:
Su, Francis E., et al. "Fermat's Last Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
