What is number theory? One might have thought that it was simply the study of numbers, but that is too broad a definition, since numbers are almost ubiquitous in mathematics. To see what distinguishes number theory from the rest of mathematics, let us look at the equation *x*^{2} + *y*^{2} = 15 925, and consider whether it has any solutions. One answer is that it certainly does: indeed, the solution set forms a circle of radius in the plane. However, a number theorist is interested in *integer* solutions, and now it is much less obvious whether any such solutions exist.

A useful first step in considering ...

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