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
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 ...