2 1. Introduction Are there any non-trivial examples? If we try to assign variables to our problem, we see that we are trying to find solutions to y2 = x(x + 1)(x + 2) (1.1) with x, y ∈ Z and y = 0. Equation (1.1) defines an elliptic curve. It turns out that there are no integral solutions other than the trivial ones (see Exercise 1.4.1). Are there rational solutions, i.e., are there solutions with x, y ∈ Q? This is a more delicate question, but the answer is still no (we will prove it in Example 2.7.6). Here is a similar question, with a very different answer: • Are there three integers that differ by 5, i.e., x, x + 5 and x + 10, and whose product is a perfect square? In this case, we are trying to find solutions to y2 = x(x+5)(x+10) with x, y ∈ Z. As in the previous example, there are trivial solutions (those which involve 0) but in this case, there are non-trivial solutions as well: (−9) · (−9 + 5) · (−9 + 10) = (−9) · (−4) · 1 = 36 = 62 40 · (40 + 5) · (40 + 10) = 40 · 45 · 50 = 90000 = 3002. Moreover, there are also rational solutions, which are far from obvious: 5 4 · 5 4 + 5 · 5 4 + 10 = 75 8 2 − 50 9 · − 50 9 + 5 · − 50 9 + 10 = 100 27 2 and, in fact, there are infinitely many rational solutions! Here are some of the x-coordinates that work: x = −9, 40, 5 4 , −50 9 , 961 144 , 7200 961 , − 12005 1681 , − 16810 2401 , − 27910089 5094049 , . . . In Sections 2.9 and 2.10 we will explain a method to find rational points on elliptic curves and, in Exercise 2.12.23, the reader will cal- culate all the rational points of y2 = x(x + 5)(x + 10). Example 1.1.2 (The Congruent Number Problem). We say that n ≥ 1 is a congruent number if there exists a right triangle whose sides are rational numbers and whose area equals n. What natural numbers are congruent?

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