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

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IV.5   Arithmetic Geometry

   Jordan S. Ellenberg

1   Diophantine Problems, Alone and in Teams

Our goal is to sketch some of the essential ideas of arithmetic geometry; we begin with a problem which, on the face of it, involves no geometry and only a bit of arithmetic.

Problem. Show that the equation

Image

has no solution in nonzero rational numbers x, y, z.

(Note that it is only in the coefficient 7 that (1) differs from the Pythagorean equation x2 + y2 = z2, which we know has infinitely many solutions. It is a feature of arithmetic geometry that modest changes of this kind can have drastic effects!)

Solution. Suppose x, y, z are rational numbers ...

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