Number Theory

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 $n \geq 3$. 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.

How have we used integer solutions to Fermat's equation in real life? Hint: they shaped history.