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 = 2^{2},
1 · 120 + 1 = 11^{2},
1 · 8 + 1 = 3^{2},
3 · 120 + 1 = 19^{2},
3 · 8 + 1 = 5^{2},
8 · 120 + 1 = 31^{2}.
{a, b, a + b + 2r, 4r(r + a)(r + b) },
where ab + 1 = r^{2}. He was also able to add the fifth positive rational,777480 / 8288641,
to the Fermat's set (see Classical references). In 2019, 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
{a_{1},
a_{2}, ... ,
a_{m}}
is called a Diophantine mtuple if

Definition 1.2: A set of m nonzero rationals
{a_{1}, a_{2}, ... ,
a_{m}}
is called a rational Diophantine mtuple if

It is natural to ask how large these sets, i.e. (rational) Diophantine tuples, can be. This question was recently 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 mtuples 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
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine mtuples
6. Connections with elliptic curves
7. Various generalizations
8. References
Diophantine mtuples page  Andrej Dujella home page 