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 $1\leq i<j\leq m$.^[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 the problem was finally resolved by He, Togbé and Ziegler.^[2]

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.^[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.

External links

• Andrej Dujella's pages on diophantine m-tuples