Diophantine quintuple

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 $1\le i < j \le m$.^[1]

Diophantus himself found the set $\left\{\frac1{16}, \frac{33}{16}, \frac{17}4, \frac{105}{16}\right\}$ of rationals which has the property that each $a_i a_j + 1$ is a rational square.^[1] More recently, sets of six positive rationals have been found.^[2]

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]

No (integer) diophantine quintuples are known, and it is an open problem whether any exist.^[1] Dujella has shown that at most a finite number of diophantine quintuples exist.^[1]

References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 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.
2. ↑ Gibbs, Philip (1999). "A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples". arXiv:math.NT/9903035v1.

External links

• Andrej Dujella's pages on diophantine m-tuples