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In mathematics, a diophantine m-tuple is a set of m positive integers such that is a perfect square for any . 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: . 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 .
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 . 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.
- Dujella, Andrej (January 2006). "There are only finitely many Diophantine quintuples". Journal für die reine und angewandte Mathematik. 2004 (566): 183–214. doi:10.1515/crll.2004.003.
- He, B.; Togbé, A.; Ziegler, V. "There is no Diophantine Quintuple". arXiv:1610.04020.
- Gibbs, Philip (1999). "A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples". arXiv:math.NT/9903035v1.
- Herrmann, E.; Pethoe, A.; Zimmer, H. G. (1999). "On Fermat's quadruple equations". Math. Sem. Univ. Hamburg. 69: 283–291. doi:10.1007/bf02940880.