In number theory, a diophantine m-tuple is a set of m positive integers
The first diophantine quadruple was found by Fermat:
[1] It was proved in 1969 by Baker and Davenport [1] that a fifth positive integer cannot be added to this set.
However, Euler was able to extend this set by adding the rational number
[1] 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.
[1] In 2016 it was shown that no such quintuples exist by He, Togbé and Ziegler.
In both of these types of extension, as for Fermat's quadruple, it is possible to add a fifth rational number rather than an integer.
[3] Diophantus himself found the rational diophantine quadruple
[1] More recently, Philip Gibbs found sets of six positive rationals with the property.
[4] 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.