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  These curves defined by very simple equations are shrouded in mystery and elegance. In fact, the equations describing them are so simple that even high-school students would be able to understand them. However, a ton of simple questions about them remain unsolved despite tenacious efforts by some of the greatest mathematicians in the world. But that’s not all. As you will soon see, this theory connects various important fields of mathematics because it turns out that elliptic curves are more than just plane curves! Let’s grab a cup of coffee and start from the beginning. Introduction and Motivation In mathematics, we often solve problems by stating them in a different setting than they originally occurred in. An example of this is that some geometric problems can be turned into algebraic problems and vice versa. A classical problem going back thousands of years is whether a positive integer  n  is the area of some right triangle with rational side lengths, that is such ...