Our purpose in this chapter is to study some elementary diophantine equations. Among these we highlight Pythagoras’ and Pell’s equation, for which we characterize all solutions. We also present to the reader the important Fermat’s descent method, which provides a frequently useful tool for showing that certain diophantine equations do not possess nontrivial solutions, in a way to be made precise. The aforementioned method is one of the major legacies of Pierre Simon de Fermat to Number Theory, and will be frequently used hereafter.
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