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?