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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