The study of Diophantine Equations, that is, equations to which we want to find integer solutions, is one of the oldest and most fruitful areas of number theory. Many famous problems, such as Fermat's Last Theorem or Catalan's Conjecture ("the only two consecutive powers are 8 and 9") are examples of Diophantine equations which have sparked great interest and, in some cases, entire areas of mathematics which have been developed in order to solve them (ring theory and elliptic curves being two notable examples).