What Are Diophantine Equations?

$$\text{Fermat's Last Theorem}$$

===

Sir Andrew Wiles just won the Abel Prize for his proof of Fermat's Last Theorem, which had remained unproven since Fermat's discovery in 1637. Fermat stated that $x^n+y^n=z^n$ does not have any integer solutions when $n\ge3$. When $n=2$, Fermat's equation simply expresses the Pythagorean Theorem. There are many solutions, in the form of Pythagorean triples. For example, $3^2+4^2=5^2$. Fermat's claim is that there are no such triples of solutions for higher powers.