Diophantine quintuple

{{short description|Set of positive integers such that the product of any two plus one is a perfect square}}

In number theory, a diophantine {{mvar|m}}-tuple is a set of {{mvar|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 $1\le i < j \le m.$ A set of {{mvar|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 {{mvar|m}}-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.{{cite journal | title = There are only finitely many Diophantine quintuples

| journal = Journal für die reine und angewandte Mathematik | last = Dujella | first = Andrej | author-link = Andrej Dujella | volume = 2004 | issue = 566 | pages = 183–214 |date=January 2006 | doi=10.1515/crll.2004.003| citeseerx = 10.1.1.58.8571 }} In 2016 it was shown that no such quintuples exist by He, Togbé and Ziegler.{{cite journal | journal = Transactions of the American Mathematical Society | title = There is no Diophantine Quintuple | arxiv = 1610.04020 | last1 = He | first1 = B. | last2 = Togbé | first2 = A. | last3 = Ziegler | first3 = V.| year = 2016 }}

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.{{cite journal|last1=Arkin|first1=Joseph|author1-link=Joseph Arkin|last2=Hoggatt|first2=V. E. Jr.|author2-link=Verner Emil Hoggatt Jr.|last3=Straus|first3=E. G.|author3-link=Ernst G. Straus|issue=4|journal=Fibonacci Quarterly|mr=550175|pages=333–339|title=On Euler's solution of a problem of Diophantus|url=https://www.fq.math.ca/Scanned/17-4/arkin.pdf|volume=17|year=1979}}

The rational case

Diophantus himself found the rational diophantine quadruple $\left\{\tfrac1{16}, \tfrac{33}{16}, \tfrac{17}4, \tfrac{105}{16}\right\}.$ More recently, Philip Gibbs found sets of six positive rationals with the property.{{cite arXiv |last= Gibbs|first=Philip |eprint= math.NT/9903035v1 |title= A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples |year= 1999}} 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.{{cite journal | title = On Fermat's quadruple equations | journal = Math. Sem. Univ. Hamburg | last1 = Herrmann | first1 = E. | last2 = Pethoe | first2 = A. | last3 = Zimmer | first3 = H. G. | volume = 69 | pages = 283–291 | date = 1999 | doi=10.1007/bf02940880| hdl = 2437/90714 | hdl-access = free }}

References

External links

[http://web.math.pmf.unizg.hr/~duje/dtuples.html Andrej Dujella's pages on diophantine m-tuples]

Category:Diophantine equations