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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