# Diophantine quintuple

In mathematics, a diophantine m-tuple is a set of m positive integers ${\displaystyle \{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\}}$ such that ${\displaystyle a_{i}a_{j}+1}$ is a perfect square for any ${\displaystyle 1\leq i.[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: ${\displaystyle \{1,3,8,120\}}$.[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 ${\displaystyle {\frac {777480}{8288641}}}$.[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 a resolution was proposed by He, Togbé and Ziegler, subject to peer-review.[2]

## The Rational Case

Diophantus himself found the rational diophantine quadruple ${\displaystyle \left\{{\frac {1}{16}},{\frac {33}{16}},{\frac {17}{4}},{\frac {105}{16}}\right\}}$.[1] More recently, Philip Gibbs found sets of six positive rationals with the property.[3] 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.[4]

## References

1. 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. ^ He, B.; Togbé, A.; Ziegler, V. "There is no Diophantine Quintuple". arXiv:1610.04020.
3. ^ Gibbs, Philip (1999). "A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples". arXiv:math.NT/9903035v1.
4. ^ Herrmann, E.; Pethoe, A.; Zimmer, H. G. (1999). "On Fermat's quadruple equations". Math. Sem. Univ. Hamburg. 69: 283–291. doi:10.1007/bf02940880.