Introduction

Here, in the midst of this sad and barren landscape of the Greek ac-

complishments in arithmetic, suddenly springs up a man with youthful

energy: Diophantus. Where does he come from, where does he go to?

Who were his predecessors, who his successors? We do not know.

It is all one big riddle. He lived in Alexandria. If a conjecture were

permitted, I would say he was not Greek; . . . if his writings were

not in Greek, no-one would ever think that they were an outgrowth

of Greek culture . . . .

Hermann Hankel (1874, translated by N. Schappacher)

Diophantine equations have a long history. More than two thousand years ago,

Diophantus of Alexandria considered, among many others, the equations

x2

+

y2

=

z2

, (∗)

y(6 − y) =

x3

− x ,

and

y2

=

x2

+

x4

+

x8

.

In Diophantus’ book Arithmetica, we find the formula

(

(p2

−

q2)λ,

2pqλ,

(p2

+

q2)λ

)

(†)

that generates infinitely many solutions of (∗). For the second and third of the

equations mentioned, Diophantus gives particular solutions, namely (1/36, 1/216)

and (1/2, 9/16), respectively.

In general, a polynomial equation in several indeterminates, where solutions are

sought in integers or rational numbers, is called a Diophantine equation in honour

of Diophantus. Diophantus himself was interested in solutions in positive integers

or positive rational numbers. Contrary to the point of view usually adopted today,

he did not accept negative numbers.

It is remarkable that algebro-geometric methods have often been fruitful in order

to understand a Diophantine equation. For example, there is a simple geometric

idea behind formula (†).

Indeed, since the equation is homogeneous, it suﬃces to look for solutions

of

X2

+ Y

2

= 1 in rationals. This equation defines the unit circle. For every t ∈ ,

there is the line “x = −ty + 1” going through the point (1, 0). An easy calculation

1

http://dx.doi.org/10.1090/surv/198/01