en>Michael Hardy at 12:39, 14 May 2013

2013-05-14T12:39:55Z

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In mathematics, a '''diophantine ''m''-tuple''' is a set of ''m'' positive integers <math>\{a_1, a_2, a_3, a_4,\ldots, a_m\}</math> such that <math>a_i a_j + 1</math> is a perfect square for any <math> 1\le i < j \le m</math>.<ref name="Dujella"/>

[[Diophantus]] himself found the set <math>\left\{\frac1{16}, \frac{33}{16}, \frac{17}4, \frac{105}{16}\right\}</math> of rationals which has the property that each <math>a_i a_j + 1</math> is a [[rational square]].<ref name="Dujella"/> More recently, sets of six positive rationals have been found.<ref name="Gibbs">{{cite arXiv |last= Gibbs|first=Philip |eprint= math.NT/9903035 |title= A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples |year= 1999 |version= v1 |accessdate= May 6, 2013 }}</ref>

The first diophantine quadruple was found by [[Pierre de Fermat|Fermat]]: <math>\{1,3, 8, 120\}</math>.<ref name="Dujella"/> It was proved in 1969 by Baker and Davenport <ref name="Dujella"/> that a fifth positive integer cannot be added to this set.
However, [[Leonhard Euler|Euler]] was able to extend this set by adding the rational number
<math>\frac{777480}{8288641}</math>.<ref name="Dujella"/>

No ([[integer]]) diophantine quintuples are known, and it is an open problem whether any exist.<ref name="Dujella"/> Dujella has shown that at most a finite number of diophantine quintuples exist.<ref name="Dujella">{{cite journal | title = There are only finitely many Diophantine quintuples
| journal = [[Journal für die reine und angewandte Mathematik]] | last = Dujella | first = Andrej | authorlink = Andrej Dujella | volume = 2004 | issue = 566 | pages = 183–214 |date=January 2006 }}</ref>

==References==
{{reflist}}

==External links==

* [http://web.math.pmf.unizg.hr/~duje/dtuples.html Andrej Dujella's pages on diophantine ''m''-tuples]

[[Category:Number theory]]

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en>Michael Hardy at 12:39, 14 May 2013