6 1. Introduction
and, if n is even,
#{(x, y, z)
Z3
:
n
2
=
4x2
+
y2
+
32z2}
=
1
2
#{(x, y, z)
Z3
:
n
2
=
4x2
+
y2
+
8z2}
.
Moreover, if the Birch and Swinnerton-Dyer conjecture is true, then,
conversely, these equalities imply that n is a congruent number.
For example, for n = 2 we have
n
2
= 1 =
4x2
+
y2
+
32z2
if and
only if x = z = 0 and y = ±1, so the left-hand side of the appropriate
equation in Tunnell’s theorem is equal to 2. However, the right-hand
side is equal to 1 and the equality does not hold. Hence, 2 is not a
congruent number.
For a complete historical overview of the congruent number prob-
lem, see [Dic05], Ch. XVI. The book [Kob93] contains a thorough
modern treatment of the problem. The reader may also find useful an
expository paper [Con08] on the congruent number problem, written
by Keith Conrad. Another neat exposition, more computational in
nature (using Sage), appears in [Ste08], Section 6.5.3.
Example 1.1.5 (Fermat’s last theorem). Let n 3. Are there any
solutions to
xn
+
yn
=
zn
in integers x, y, z with xyz = 0? The
answer is no. In 1637, Pierre de Fermat wrote in the margin of a
book (Diophantus’ Arithmetica; see Figure 9 in Section 5.5) that he
had found a marvellous proof, but the margin was too small to contain
it. Since then, many mathematicians tried in vain to demonstrate (or
disprove!) this claim. A proof was finally found in 1995 by Andrew
Wiles ([Wil95]). We shall discuss the proof in some more detail
in Section 5.5. For now, we will outline the basic structure of the
argument.
First, it is easy to show that, to prove the theorem, it suffices to
show the cases n = 4 and n = p 3, a prime. It is not difficult to
show that
x4 +y4
=
z4
has no non-trivial solutions in Z (this was first
shown by Fermat). Now, suppose that p 3 and a, b, c are integers
with abc = 0 and ap + bp = cp. Gerhard Frey conjectured that if such
a triple of integers exists, then the elliptic curve
E :
y2
= x(x
ap)(x
+
bp)
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