1.1. Elliptic curves 5 Figure 2. Two rational points on the curve y2 = x3 − 25x. congruent number. For example, n = 157 ≡ 5 mod 8 must be a con- gruent number and, indeed, Don Zagier has exhibited a right triangle (a, b, c) whose area equals 157. The hypotenuse of the simplest such triangle is: c = 2244035177043369699245575130906674863160948472041 8912332268928859588025535178967163570016480830 . In Example 5.2.7 we will see an application of the conjecture of Birch and Swinnerton-Dyer to find a rational point P on y2 = x3 − 1572x, which corresponds to a right triangle of area 157 via the correspon- dence in Proposition 1.1.3. The best known result on the congruent number problem is due to J. Tunnell: Theorem 1.1.4 (Tunnell, 1983, [Tun83]). If n is an odd square- free positive integer and n is the area of a right triangle with rational sides, then the following cardinalities are equal: #{(x, y, z) ∈ Z3 : n = 2x2 + y2 + 32z2} = 1 2 ( # (x, y, z) ∈ Z3 : n = 2x2 + y2 + 8z2 )

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