Diophantine quintuple explained

\{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. A set of 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 -tuple.

Diophantine m-tuples

The first diophantine quadruple was found by Fermat:

\{ 1 , 3 , 8 , 120 \}. It was proved in 1969 by Baker and Davenport that a fifth positive integer cannot be added to this set.However, Euler was able to extend this set by adding the rational number\tfrac{ 777480 }{ 8288641 }.

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\{\tfrac 1 { 16 }, \tfrac{ 33 }{ 16 }, \tfrac{ 17 } 4 , \tfrac{ 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]

External links

• Andrej Dujella's pages on diophantine m-tuples

Notes and References

1. There are only finitely many Diophantine quintuples . . Dujella . Andrej . Andrej Dujella . 2004 . 566 . 183–214 . January 2006 . 10.1515/crll.2004.003. 10.1.1.58.8571 .
2. . There is no Diophantine Quintuple . 1610.04020 . He . B. . Togbé . A. . Ziegler . V.. 2016 .
3. Arkin. Joseph. Joseph Arkin. Hoggatt. V. E. Jr.. Verner Emil Hoggatt Jr.. Straus. E. G.. Ernst G. Straus. 4. Fibonacci Quarterly. 550175. 333–339. On Euler's solution of a problem of Diophantus. 17. 1979.
4. Gibbs. Philip . math.NT/9903035v1 . A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples . 1999.
5. On Fermat's quadruple equations . Math. Sem. Univ. Hamburg . Herrmann . E. . Pethoe . A. . Zimmer . H. G. . 69 . 283–291 . 1999 . 10.1007/bf02940880.