Some classical diophantine examples

Abstract

In this elementary chapter we shall deal with some classical diophantine equations, to be solved in ordinary integers of ℤ. After a brief study of the case of a single variable and of the linear case, we shall go to quadratic equations in two variables, which represent conics in A². The fundamental theory here comes from the Pell Equation X ² − dY ² = 1, where d is a fixed positive integer, not a square. This study also links diophantine equations with diophantine approximation, a theory which provides most important tools, that we shall meet throughout. After Pell Equation we shall give a complete effective treatment of the integral points for general conics, i.e. quadratic equations in two variables to be solved in ℤ².