# Diophantine quintuple

In mathematics, a diophantine m-tuple is a set of m positive integers $\{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\}$ such that $a_{i}a_{j}+1$ is a perfect square for any $1\leq i . 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: $\{1,3,8,120\}$ . 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 ${\frac {777480}{8288641}}$ .

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 a resolution was proposed by He, Togbé and Ziegler, subject to peer-review.

## The Rational Case

Diophantus himself found the rational diophantine quadruple $\left\{{\frac {1}{16}},{\frac {33}{16}},{\frac {17}{4}},{\frac {105}{16}}\right\}$ . 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.