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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.2 Analytic Number Theory

Andrew Granville

1 Introduction

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 x2 + y2 = 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 Image 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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