xⁿ + yⁿ = zⁿJust look at that equation (more at: MathWorld).
At first glance, most of us would probably see 3² + 4² = 5², which is an example of Pythagoras' Theorem.
But for cases where the integer n = 2 or greater, are there any solutions? (By the way, x, y, and z are nonzero integers.)
A 17th-century French lawyer (and mathematician) named Fermat said that there aren't. This statement became known as Fermat's Last Theorem.
It was proven true only hundreds of years later, around 1993 to 1995 by Andrew Wiles, a British mathematician.
When asked whether his proof is the same as Fermat's, Wiles responded:
There's no chance of that. Fermat couldn't possibly have had this proof. It's 150 pages long. It's a 20th-century proof. It couldn't have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren't around in Fermat's time.I took a look at the mathematics being used in the proof, and it made me dumbstruck. There was just too much math jargon that I didn't bother comprehending the material fully.
The few new words that got stuck in my mind are elliptic curves, Taniyama-Shimura conjecture, and Galois representations, although I know _nothing_ about them.
To be continued.