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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.
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showed that at most a finite number of diophantine quintuples exist. In 2016 it was shown that no such quintuples exist by He, Togbé and
Ziegler.
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More recently, Philip Gibbs found sets of six positive rationals with the property. It is not known whether any larger rational diophantine
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It was proved in 1969 by Baker and
Davenport that a fifth positive integer cannot be added to this set. However,
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Gibbs, Philip (1999). "A Generalised Stern-Brocot Tree from
Regular Diophantine Quadruples".
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Set of positive integers such that the product of any two plus one is a perfect square
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Herrmann, E.; Pethoe, A.; Zimmer, H. G. (1999). "On Fermat's quadruple equations".
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349:) diophantine quintuples was one of the oldest outstanding unsolved problems in
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He, B.; Togbé, A.; Ziegler, V. (2016). "There is no
Diophantine Quintuple".
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494:(January 2006). "There are only finitely many Diophantine quintuples".
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was able to extend this set by adding the rational number
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336:{\displaystyle {\tfrac {777480}{8288641}}.}
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370:Diophantus
353:. In 2004
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220:A set of
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672:Category
571:(1979).
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663:-tuples
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347:integer
325:8288641
252:-tuples
230:product
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322:777480
258:Fermat
242:-tuple
29:-tuple
609:arXiv
576:(PDF)
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309:Euler
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