## 1. Introduction

The Greek mathematician Diophantus of Alexandria first studied the problem of finding four numbers such that the product of any two of them increased by unity is a perfect square. He found a set of four positive rationals with the this property:

{1/16, 33/16, 17/4, 105/16}.

However, the first set of four positive integers with the above property,

{1, 3, 8, 120},

was found by Fermat. Indeed, we have

1 · 3 + 1 = 22,         1 · 120 + 1 = 112,
1 · 8 + 1 = 32,         3 · 120 + 1 = 192,
3 · 8 + 1 = 52,         8 · 120 + 1 = 312.

Euler found the infinite family of such sets:

{a, b, a + b + 2r, 4r(r + a)(r + b) },

where ab + 1 = r2. He was also able to add the fifth positive rational,

777480 / 8288641,

to the Fermat's set (see Classical references). In 2017, Stoll [372] proved that extension of Fermat's set to a rational quintuple with the same property is unique.

In January 1999, the first example of a set of six positive rationals with the property of Diophantus and Fermat was found by Gibbs [144, 80]:

{11/192, 35/192, 155/27, 512/27, 1235/48, 180873/16}.

These examples motivate the following definitions.

 Definition 1.1: A set of m positive integers {a1, a2, ... , am} is called a Diophantine m-tuple if ai · aj + 1 is a perfect square for all 1 ≤ i < j ≤ m.

 Definition 1.2: A set of m non-zero rationals {a1, a2, ... , am} is called a rational Diophantine m-tuple if ai · aj + 1 is a perfect square for all 1 ≤ i < j ≤ m.

It is natural to ask how large these sets, i.e. (rational) Diophantine tuples, can be. This question was recently almost completely solved in the integer case (see Chapter 2). On the other hand, it seems that in the rational case we do not have even a widely accepted conjecture. In particular, no absolute upper bound for the size of rational Diophantine m-tuples is known.

In the integer case we have the following folklore "Diophantine quintuple conjecture".

 Conjecture 1.1: There does not exist a Diophantine quintuple.

The first important result concerning this conjecture was proved in 1969 by Baker and Davenport [3]. Using Baker's theory on linear forms in logarithms of algebraic numbers and a reduction method based on continued fractions, they proved that if d is a positive integer such that {1, 3, 8, d} forms a Diophantine quadruple, then d = 120. It implies that the Fermat's set {1, 3, 8, 120} cannot be extended to a Diophantine quintuple. This problem was stated in 1967 by Gardner [1] and in 1968 by van Lint [2]. The same result was proved later, with different methods, by Kanagasabapathy & Ponnudurai [4], Sansone [6] and Grinstead [11].

2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine m-tuples
6. Connections with elliptic curves
7. Various generalizations
8. References