Diophantine quintuple

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

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 ${\displaystyle \frac{777480}{8288641}.}$^[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.Cite error: Closing </ref> missing for <ref> tag

Diophantus himself found the rational diophantine quadruple ${\displaystyle \{\frac{1}{16},\frac{33}{16},\frac{17}{4},\frac{105}{16}\}.}$^[1] More recently, Philip Gibbs found sets of six positive rationals with the property.^[2] 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.^[3]

• Andrej Dujella's pages on diophantine m-tuples

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Diophantine quintuple. Read more