4 1. Introduction n = 5, and Fibonacci found the triangle ( 3 2 , 20 3 , 41 6 ). We will explain a method to find this triangle below. In 1225, Fibonacci wrote a more general treatment about the congruent number problem, in which he stated (without proof) that if n is a perfect square, then n cannot be a congruent number. The proof of such a claim had to wait until Pierre de Fermat (1601-1665) settled that the number 1 (and every square number) is not a congruent number (a result that he showed in order to prove the case n = 4 of Fermat’s last theorem). The connection between the congruent number problem and el- liptic curves is as follows: Proposition 1.1.3. The number n 0 is congruent if and only if the curve y2 = x3 n2x has a point (x, y) with x, y Q and y = 0. More precisely, there is a one-to-one correspondence Cn ←→ En between the following two sets: Cn = {(a, b, c) : a2 + b2 = c2, ab 2 = n} En = {(x, y) : y2 = x3 n2x, y = 0}. Mutually inverse correspondences f : Cn En and g : En Cn are given by f((a, b, c)) = nb c a , 2n2 c a , g((x, y)) = x2 n2 y , 2nx y , x2 + n2 y . The reader can provide a proof (see Exercise 1.4.3). For example, the curve E : y2 = x3 25x has a point (−4,6) that corresponds to the triangle ( 3 2 , 20 3 , 41 6 ). But E has other points, such as ( 1681 144 , 62279 1728 ) that corresponds to the triangle 1519 492 , 4920 1519 , 3344161 747348 which also has area equal to 5. See Figure 2. Today, there are partial results toward the solution of the congru- ent number problem, and strong results that rely heavily on famous (and widely accepted) conjectures, but we do not have a full answer yet. For instance, in 1975 (see [Ste75]), Stephens showed that the Birch and Swinnerton-Dyer conjecture (which we will discuss in Sec- tion 5.2) implies that any positive integer n 5, 6 or 7 mod 8 is a
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