Diophantine quintuple

Short description: Set of positive integers such that the product of any two plus one is a perfect square

In number theory, a diophantine m-tuple is a set of m positive integers $\{a_1,a_2,a_3,a_4,\dots ,a_m\}$ such that $a_i{a}_j+1$ is a perfect square for any $1\le i<j\le m.$ 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.

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.

The rational case

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

External links

• Andrej Dujella's pages on diophantine m-tuples