Fermat's Last Theorem

"While reading his Latin translation of Diophantus's Greek masterpiece Arithmetica, he [Fermat] wrote a deceptively simple comment in Latin next to a problem about finding squares that are sums of two other squares (for example, 3^2 + 4^2 = 5^2). The comment he made is now known as Fermat's Last Theorem ("F.L.T." to its closest friends). Scientific American has aptly translated it into English:

'On the other hand, it is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.'

Restated, the equation x^n + y^n = z^n, in which n is an integer greater than 2, has no solution in positive integers. This means that there are not positive whole numbers that solve the equation when the exponent n is greater than 2. When n is 2, there is an infinity of solutions. One such solution is the Pythagorean theorem, which states that the sum of the squares of the lengths of two sides of a right-angled triangle is equal to the square of the length of the hypotenuse...

Unfortunately, Fermat died without ever offering a proof to the world of mathematics, and people have been searching for one ever since."

Marilyn Vos Savant The World's Most Famous Math Problem