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## What is a Mathematical Proof? An Introduction to what the entire field of mathematics relies upon...

## Fermat's Last Theorem

One of my favorite math stories is that of the proof of Fermat's Last Theorem. Here is some background about the problem:

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

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

The documentary about Andrew Wiles is on www.youtube.com and is worth the time. He is such a modest man who spent his entire career struggling with this proof.

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"!The documentary about Andrew Wiles is on www.youtube.com and is worth the time. He is such a modest man who spent his entire career struggling with this proof.