Fermat's Last Theorem
Fermat's last theorem is particularly famous in number theory. It was proposed by Fermat in the margin of his book Arithmetica. The note in the margin (when translated) read: "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." However, nobody was able to prove this theorem for centuries. Finally, in 1993 Andrew Wiles produced a proof of the theorem - a proof that was much more complicated than anything Fermat could have produced himself.
Given integers and , with there are no solutions to the equation: