# Fermat’s Last Theorem

There are lots of Pythagorean triples; triples of whole numbers which satisfy:
x2 + y2 = z2. But are there any which satisfy
xn + yn = zn, 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 Taniyama-Shimura conjecture in arithmetic/algebraic geometry implies Fermat’s Last Theorem. Wiles was able to prove the Taniyama-Shimura 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 centuries-old 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.

Su, Francis E., et al. “Fermat’s Last Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
H. Edwards, Fermat’s Last Theorem, Springer-Verlag, 1977.
Barry Cipra, What’s Happening in the Mathematical Sciences, 1995-1996. Amer. Math. Soc., 1996.

Fun Fact suggested by:
Francis Su

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