Here we will introduce some of the leading characters in our story. To give some form to our tale, we will consider a particular question: Why were Greek mathematicians able to solve certain kinds of algebraic problems and not others? Of course, the Greeks had no concept corresponding exactly to our “algebra,” but we feel free to be anachronistic and to use modern algebraic equations to describe what they were doing. The various concepts and terms we use in the Prologue will be explained in greater detail later in the book.

To start with, let’s define an “elliptic curve” as an equation of the form

*y*^{2} = *x*^{3} + *Ax* + *B*

where *A* and *B* are fixed ...

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