Abstract: Rational points on elliptic curves play a very important role in number theory and arithmetic algebraic geometry. They lead us immediately to a variety of difficult results and conjectures, for example, the modularity theorem of Breuil-Conrad-Diamond-Taylor-Wiles and the Birch and Swinnerton-Dyer conjecture. This may not sound like an elementary talk. But fortunately, we can introduce these results in a very elementary way, by considering an old question on rational right triangles: which positive integers can be the area of a right triangle with rational sides ?