In mathematics, a diophantine m-tuple is a set of m positive integers
{
a
1
,
a
2
,
a
3
,
a
4
,
…
,
a
m
}
such that
a
i
a
j
+
1
is a perfect square for any
1
≤
i
<
j
≤
m
. A set of m positive rational numbers with the similar property that the product of any two is one less than a rational square is known as a rational diophantine m-tuple.
The first diophantine quadruple was found by Fermat:
{
1
,
3
,
8
,
120
}
. It was proved in 1969 by Baker and Davenport that a fifth positive integer cannot be added to this set. However, Euler was able to extend this set by adding the rational number
777480
8288641
.
The question of existence of (integer) diophantine quintuples was one of the oldest outstanding unsolved problems in Number Theory. In 2004 Andrej Dujella showed that at most a finite number of diophantine quintuples exist. In 2016 the problem was finally resolved by He, Togbé and Ziegler.
Diophantus himself found the rational diophantine quadruple
{
1
16
,
33
16
,
17
4
,
105
16
}
. More recently, Philip Gibbs found sets of six positive rationals with the property. It is not known whether any larger rational diophantine m-tuples exist or even if there is an upper bound, but it is known that no infinite set of rationals with the property exists.
