Fermat's last theorem

In Mathematics, Fermat's last theorem is the statement

The sum of like powers of two nonzero integers will never be another such power when the power is greater than 2

Using algebraic notation, it is the statement that for each fixed power $n\geq 3$, it is impossible to find a trio of nonzero integers $x,y,{\text{ and }}z$ satisfying the equation

$x^{n}+y^{n}=z^{n}$

This statement was first recorded by Pierre de Fermat in the margin of his copy of the treatise Arithmetica by Diophantus. He then claimed to have discovered a marvelous proof which, unfortunately, was too long for the margin to contain. This became the most tantalizing mathematical claim in history, spurring generations of mathematicians to unsuccessful attempts at providing a rigorous proof. For centuries, only halting progress was made, through the efforts of some of history's most brilliant mathematicians, including Pierre de Fermat, Leonhard Euler, Carl Friedrich Gauss, and Ernst Eduard Kummer. Fermat's last theorem was finally proved by Andrew Wiles in 1995, instantly making him the most famous mathematician alive today.

Fermat's last theorem has no scientific application, nor has it found use as a tool for proving other mathematical statements. Its apparent lack of utility notwithstanding, it is one of the most important mathematical results in history. Attempts at proving the theorem directly resulted in the development of important mathematical theories that 'do' have important applications to other fields. These include the unique factorization theorem, algebraic number fields, elliptic curves, and modular forms.

The theorem also has had a broader impact as a cultural ambassador for mathematics. Its statement can be understood by a child, and until recently, was one of very few such simple claims which even experts could not verify. The announcement of its proof received more media coverage than any mathematical result in history, including world-wide newspaper coverage and BBC Horizon and PBS NOVA documentaries.