In mathematics, a diophantine m-tuple is a set of m positive integers \( \{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\} \) such that \( a_{i}a_{j}+1 \) is a perfect square for any \( 1\leq i<j\leq m \).[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.
Diophantine m-tuples
The first diophantine quadruple was found by Fermat: \( \{1,3,8,120\} \).[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 \( {\frac {777480}{8288641}} \).[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.[1] In 2016 it was shown that no such quintuples exist by He, Togbé and Ziegler.[2]
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.[3]
The rational case
Diophantus himself found the rational diophantine quadruple \( \left\{{\frac 1{16}},{\frac {33}{16}},{\frac {17}4},{\frac {105}{16}}\right\} \) .[1] More recently, Philip Gibbs found sets of six positive rationals with the property.[4] 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.[5]
References
Dujella, Andrej (January 2006). "There are only finitely many Diophantine quintuples". Journal für die reine und angewandte Mathematik. 2004 (566): 183–214. CiteSeerX 10.1.1.58.8571. doi:10.1515/crll.2004.003.
He, B.; Togbé, A.; Ziegler, V. (2016). "There is no Diophantine Quintuple". Transactions of the American Mathematical Society. arXiv:1610.04020.
Arkin, Joseph; Hoggatt, V. E., Jr.; Straus, E. G. (1979). "On Euler's solution of a problem of Diophantus" (PDF). Fibonacci Quarterly. 17 (4): 333–339. MR 0550175.
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.
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