Diophantine quintuple
In number theory, a diophantine m-tuple is a set of m positive integers [math]\displaystyle{ \{a_1, a_2, a_3, a_4,\ldots, a_m\} }[/math] such that [math]\displaystyle{ a_i a_j + 1 }[/math] is a perfect square for any [math]\displaystyle{ 1\le i \lt j \le m. }[/math][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: [math]\displaystyle{ \{1,3, 8, 120\}. }[/math][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 [math]\displaystyle{ \tfrac{777480}{8288641}. }[/math][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
The rational case
Diophantus himself found the rational diophantine quadruple [math]\displaystyle{ \left\{\tfrac1{16}, \tfrac{33}{16}, \tfrac{17}4, \tfrac{105}{16}\right\}. }[/math][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]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 Cite error: Invalid
<ref>tag; no text was provided for refs namedDujella - ↑ 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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