Fermat's Last Theorem was the last equation in a book written by Pierre de Fermat's that was the last to be solved. The equation was x^n+y^n=z^n. Pierre said that he had proof that this equation could never be proven if n was larger than 2.
He wrote this in 1637 and it hasn't been proven until 1993(1995 for perfected) by Andrew Wiles. Andrew proved this after working on the equation for 7 years. Solving it was a dream of his since he was a young boy. Andrew received worldwide recognition for his proof. Andrew solved this by also proving the Taniyama-Shimura Conjecture, which states that every elliptic curve is also modular. Andrew solved this by turning the elliptic curves into Galois representations and turning the equation into a class number formula. Many had tried before Andrew but none succeeded for 300 years.
Many doubt if Fermat had any real proof but it was still a mathematical marvel of a challenge and we can hope another such equation will pop up.
"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." - Pierre de Fermat
Every mathematician hates and loves Andrew Wiles for his proof of Fermat's Last Theorem