Number Theory/Diophantine Equations

Introduction

The theory of Diophantine equations is an ancient subject that typically involves solving (a system of) polynomial equation in integers. Perhaps the most famous diophantine equation is Fermat's, who stated in the 17th century that the equation $x^{n}+y^{n}=z^{n}$ has no solution in integers if $n$ is greater than or equal to 3 (this is called Fermat's last theorem). Fermat never published his solution to this problem (that he actually had a valid solution is highly debated)and it took mathematicians the next 350 years to finally solve this problem.