Diophantine quintuple

In number theory, a diophantine m-tuple is a set of m positive integers ${\displaystyle \{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\}}$ such that ${\displaystyle a_{i}a_{j}+1}$ is a perfect square for any ${\displaystyle 1\leq i<j\leq m.}$^[1] 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.

Diophantine m-tuples

The first diophantine quadruple was found by Fermat: ${\displaystyle \{1,3,8,120\}.}$^[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 ${\displaystyle {\tfrac {777480}{8288641}}.}$^[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.^[2]

As Euler proved, every Diophantine pair can be extended to a Diophantine quadruple. The same is true for every Diophantine triple. 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]

The rational case

Diophantus himself found the rational diophantine quadruple ${\displaystyle \left\{{\tfrac {1}{16}},{\tfrac {33}{16}},{\tfrac {17}{4}},{\tfrac {105}{16}}\right\}.}$^[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.^[5]